DESIGN AND ANALYSIS OF A MICROMECHANICAL
TUNING FORK GYROSCOPE
by
David W. King
B.S. Mechanical Engineering, Lehigh University
(1987)
SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN
AERONAUTICS AND ASTRONAUTICS
at the
©
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May, 1989
Signature of Author
I-Department of A06nautics and Astronautics
"%'y, 1989
Approved by
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Burton Boxenhorn
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DESIGN AND ANALYSIS OF A MICROMECHANICAL TUNING FORK GYROSCOPE
by
David W. King
Submitted to the Department of Aeronautics and Astronautics on May 12, 1989 in
partial fulfillment of the requirements for the degree of Master of Science in
Aeronautics and Astronautics
ABSTRACT
This thesis investigates the feasibility of a micromechanical tuning fork as an angular
rate sensor. The gyroscopic effect of a tuning fork, which is caused by the oscillating mass
moment of inertia due to the vibrating tines, is described and demonstrated by a simple
model. The gyroscopic response to an input rate about the longitudinal axis between the
tines is described by a second order, linear periodic (LP) differential equation which is
derived by Hamilton's variational method. If the periodic terms in the equation of motion are
neglected, the gyroscopic output is shown to be oscillatory in phase with the vibrating tines
and linearly related to the input rate, the system quality factor, and the mechanical gain of the
gyroscope. The mechanical gain is the ratio of the oscillating component of the mass moment
of inertia about the input axis to the nominal component.
The periodic terms in the equation of motion are not negligable for a lightly damped,
high frequency micromechanical system. A sufficient condition for stability of the LP system
is derived from the properties of the Mathieu Equation. The stability condition is specified in
terms of the product of the system quality factor and the mechanical gain, and it is checked by
applying Floquet Theory. The response of the gyroscopically forced LP equation is solved
numerically and also estimated by a Fourier Series solution. Both of these solutions
correspond with the estimated linear response within the stability region.
Mathematical models are derived for the electrostatic driving force between the two
fork tines, the variable capacitance sensing of the gyroscopic rotation, the stiffness properties
of the structure, and the air damping torque. In addition, possible error sources are analyzed
including cross-axis sensitivity, external forces and vibration, unbalance torques, motorsensor coupling, amplifier noise, and Brownian Noise. The system mathematical model is
implemented into a computer program and a baseline design configuration is obtained which
is compatible with micromachining processes. The predicted performance of the baseline
design is shown to be competitive with the double gimbal micromechanical gyroscope
currently being developed.
Thesis Supervisor:
Title:
Dr. Walter M. Hollister
Professor of Aeronautics and Astronautics
ACKNOWLEDGEMENTS
I wish to thank Burt Boxenhorn of the Charles Stark Draper Laboratory who is
responsible for the original idea of a micromechanical tuning fork gyroscope, and who has
stimulated many of the ideas presented in this paper through his pioneering work on
micromechanical inertial instruments. I must also extend thanks to my advisor, Professor
Hollister, who provided invaluable guidance and support throughout this project. Also, I
would like to express gratitiude to my colleagues who offered assistance, with special thanks
to John Connally, Doug Loose, and Norm Wereley. Finally, I am indebted to The Charles
Stark Draper Laboratory for providing generous support of the Micromechanical Instruments
project.
This report was prepared at the Charles Stark Draper Laboratory, Inc. under internal
funding.
Publication of this report does not constitute approval by the Draper Laboratory or the
sponsoring agency of the findings or conclusions contained herein.
I hereby assign my copyright of this thesis toThe Charles Stark Draper
Laboratory, Inc., Cambridge, Massachusetts.
David W. King
Permission is hereby granted by The Charles Stark Draper Laboratory, Inc. to the
Massachusetts Institute of Technology to reproduce any or all of this thesis.
TABLE OF CONTENTS
LIST OF SYMBOLS ................................................ vi
INTRODUCTION.........................................................................1
1
GYROSCOPE DESCRIPTION AND SYSTEM MODEL................................5
1.1
Tuning Fork Gyroscopic Effect.............................................
5
1.2
Demonstration Model........................................................... 8
1.3
Micromechanical Configuration...........................
........... 13
2
D Y N A M IC S ...................................
............................................
2.1
System Model...............................................................
2.2
Neglected M odes....................................................................
2.3
Equations of M otion...............................................................
2.4
Simplified Output Solution.................................
..........
2.5
Simplified Output Transient Response......................................
2.6
Cross Axis Terms ....................................................
3
SYSTEM RESPONSE ANALYSIS................................
........... 33
3.1
Periodically Time-Varying System ............................................. 33
3.2
Stability in Terms of the Mathieu Equation.............................
. 36
3.3
Approximate Forced System Response by Harmonic Balance............. 44
3.4
Numerical Solution ...................................................
...... 51
3.5
Stability in Terms of Floquet Theory................................... 63
4
SYSTEM MATHEMATICAL MODEL................................
.......
4.1
Motor Model ................................................
4.2
Sensor M odel ........................................................................
4.3
Mechanical Factors ....................................
...............
4.4
Dam ping M odel.....................................................................
4.5
Error Sources...............................................
...................
5
D E SIG N .....................................................................................
5.1
Geometric Constraints .............................
5.2
Performance Tradeoffs ........................................
16
16
19
24
27
29
30
66
68
81
84
88
95
105
106
109
6
CONCLUSIONS ...........................................................................
Comparison with Current Micromechanical Gyro Configuration.........
6.1
Comparison with Previously Developed Tuning Fork Gyros................
6.2
6.3
Future Research ................................
116
118
120
122
REFERENCES ............................................................................. 125
APPENDIX ............................................................................
..
127
LIST OF SYMBOLS
ly
10
Io
kt
bt
Wx (Ch.2)
Wz (Ch.2)
ton
(0
19bi
ifb
I
Ix
Iz
Alz
Ix,y,-:Iz
6x'1 iy',11z
E
G
L (Lt, 1)
h
w
If , fy
fz (2b)
fx (2a)
mf
mw
mp
mt
principal mass moment of inertia about sensitive axis
nominal ( non-oscillating ) component of Iy
oscillating component of Iy
rotation angle of the fork relative to its base
linear stiffness coefficient of torsional spring
linear damping coefficient of torsional system
applied angular rate about sensitive axis
applied angular rate about x-axis
applied angular rate about z-axis
motor (tines) driving frequency
natural frequency of torsional system
system frequency when od is tuned to on
angular velocity vector of fork base with respect to inertial space
angular velocity vector of fork with respect to its base
angular velocity vector of fork with respect to inertial space
inertia tensor
principal mass moment of inertia about x-axis
principal mass moment of inertia about z-axis
oscillating component of Iz
unit vectors corresponding to fork axes
unit vectors corresponding to fork base axes
modulus of elasticity for silicon
torsional modulus of elasticity for silicon
length of tines (y-dim)
height of tines (z-dim)
width of tines (x-dim)
length of flexures
thickness of flexures
width of flexures
mass of each flexure
mass of each wing
mass of each inertial proof mass
mass of each tine
Wx, Wy, wz
Q
a,q
a (t)
b (t)
c (t)
d (t)
G (t)
B (t)
an
bn
x
F1
F2
F0
eO
d
g
Vd
V0
w (z)
K (k)
E (k)
sn (x)
rOm
rot
xl
x2
Ci
Cd
CO
AC
Cfb
Rfb
x,y,z dimensions of each wing, respectively
system quality factor
gyroscope mechanical gain
Mathieu Equation parameters
second derivative term coefficient to 2nd order LP equation
first derivative term coefficient to 2nd order LP equation
first order term coefficient to 2nd order LP equation
forcing function term for 2nd order LP equation
system plant matrix
system forcing matrix
Fourier sine coefficients
Fourier cosine coefficients
state transition matrix
second order system state vector
parallel electric flux motor force
fringing electric flux motor force
total motor force
permittivity constant
nominal distance between tines
nominal distance between wing and bridge electrode
differential driving voltage
voltage with respect to ground applied on each tine
conformal transformation function
complete elliptic integral of first kind
complete elliptic integral of second kind
Jacobi's elliptic sine function
nominal distance between inertial mass center and sensitive axis
nominal distance between tine center and sensitive axis
distance from sensitive axis to front edge of sensor plate
distance from sensitive axis to back edge of sensor plate
amount of sensor capacitance increase due to rotation
amount of sensor capacitance decrease due to rotation
nominal capacitance of each pair of sensor plates
net change in capacitance output signal
output amplifier feedback capacitance
output amplifier feedback resistance
vii
Ox
sensor excitation voltage
sensor excitation frequency
B
bandwidth of output signal filter
eout
output voltage
bx, by, bz
x,y,z dimensions of strain beam slot, respectively
boron-silicon misfit factor
4Isi
Poisson's ratio for silicon
a0
axial prestress on flexures
U00
maximum free stream air velocity normal to wings
viscosity of air flow over wings
Ex
QU
volumetric flow rate through sensor channel
Qf'
profile flow rate through sensor channel
Ff
damping force due to flow impingement
Fp
Az
damping force due to pumping effect
amplitude of tine vibration at tine midpoint
z-direction misalignment of proof masses
AB
amplitude of white Brownian Noise
Aa
amplitude of white amplifier noise
a (BW)
b
system bandwidth
Ke
electronics constant
Boltzmann's constant
xm
KB
K0
dynamics constant ( 10 o)
ass,B
capacitance constant
output standard deviation due to Brownian Noise
ass,a
output standard deviation due to amplifier noise
K1
sensor excitation constant
mx, my, mz
x,y,z dimensions of each proof mass, respectively
u2
ratio of tine vibration amplitude to its maximum limit
output oscillation amplitude in quadrature due to unbalance
0quad
viii
INTRODUCTION
During the past several years, rapid advancements have been made in the fields of
integrated circuits and microchip processing. The efficiency and accuracy in which silicon
can be etched into tiny structures with dimensions in microns has led scientists to pursue the
possibility of producing mechanical elements on the microchip level. Devices such as
accelerometers, pressure transducers, flow sensors, force transducers, electrostatic actuators,
and miniature microphones have already been developed and are beginning to become
commercially available. The advantages of micromechanical elements are obvious. The
small size of the devices open up areas of application previously deemed unrealistic. For
example, micromechanical sensors are sought after for various biomedical purposes in which
conventional sensors are ineffective due to size restrictions. Also, the semiconductor batch
processing techniques applied to mechanical structures greatly reduces the cost for mass
production. It has now become vital for sophisticated sensors to become available cheaper in
order to keep in pace with the rapidly decreasing cost of microprocessors. The two
fundamental properties of silicon make it an excellent material for sensing instruments. Its
semiconductor electrical properties allow easy integration with system electronics, and its
strong mechanical properties allow micromechanical instruments to be safely operated in high
impact or acceleration environments.
Following this trend, in 1983 the Charles Stark Draper Laboratory, Inc. initiated the
design and development of a novel micromechanical gyroscope. This represented the first
development of its kind and led to a full scale project with the final goal being a complete
three axis micromechanical inertial guidance system. It has been forecast that the system will
be operated in a strapdown mode with on-chip electronics, and only consist of one square
inch flatpack. The gyroscope, designed by B. Boxenhorn [1,2] is planar, etched from a
silicon wafer, and uses an oscillating angular momentum vector to sense angular rate about
the axis normal to the gyro plane. A vibrating outer gimbal, supported by torsional pivots,
vibrates sinusoidally driven by electrostatic torquers. The oscillating angular momentum
vector, combined with the angular rate, produces a gyroscopic torque on an inner gimbal
causing it to vibrate about the other planar axis at the driving frequency. The inner gimbal
has a vertical bar mounted on it to add inertia to the gyroscopic element. The amplitude of the
inner gimbal vibration is proportional to the applied angular rate and is sensed by
ultrasensitive variable capacitance plates. The driving and sensing of the device is done by
fixing electrodes above the gimbals and applying appropriate voltages.
The gyroscope has been extensively modeled, built, and tested. On 1 September
1987, The first ever micromechanical gyroscope was shown to produce gyroscopic action.
Since that important milestone rapid improvements have been made on the device including
the design of a closed loop rebalance system, improvements in fabrication techniques making
possible a monolithic structure with buried electrodes, and analytical models of the noise
sources. The most recent test results, operated closed loop, show a gyro sensitivity of 3.13
mV/rad/sec with a noise level equivalent to 2500 degrees per hour. The gyro is projected to
achieve angular rate sensing accuracy within 10 degrees per hour upon completion of the
development program. At that point the device is expected to be useful in applications such
as small projectile guidance, kinetic energy weapon guidance and control, land vehicle
guidance and control, flight vehicle structure control and sensor stabilization, and robotics.
These applications exploit the micromechanical gyroscope's qualities not present in
conventional gyros such as small size, low cost, and ruggedness.
The idea of a vibratory gyroscope is not new. In the 1950's extensive research was
performed on vibratory angular rate sensors. The most popular configuration analyzed was
that of a tuning fork, where the two fork tines are forced to vibrate aganinst each other to give
an oscillating mass moment of inertia about the input axis. This, in turn, produces an
oscillating torque about the axis of the fork stem proportional to an applied angular about the
same axis. The Sperry Gyroscope Company designed and produced a class of these
instruments labeled the "Gyrotron Angular Rate Tachometer" [3]. Sperry envisioned that the
Gyrotron would become a high performance gyro suitable for long range navigation
applications. The Gyrotron, they believed, would acquire very high accuracy through
continued research on eliminating the zero rate error and further modification of the angular
displacement sensing technique. A decade of research continued, but, as seen retrospectively
thirty years later, the tuning fork gyro proved unsuccessful and its development was
terminated.
However, it has become desirable to investigate the potential of a tuning fork
configuration on a micromechanical level, and that is the basis of this thesis. Several factors
inspired the study of an alternate configuration micromechanical gyroscope, and they are
listed below:
1) An investigation of the feasibility of producing a micromechanical gyroscope with
an input axis on the same plane as the gyro. A combination of two of these gyros
with the current gyro would enable three gyros on a chip to sense angular rates about
all three axes.
2) A thorough analysis of the tuning fork dynamics. The previous analysis of
tuning fork gyros neglects the time variant terms that appear in the equations of
motion. On a micromechanical level these terms become significant since it is a
lightly damped system of high frequency. It is thus necessary to determine the effect
of these terms on the gyro response and if they lead to regions of instability and
poorly behaved output.
3) A simple but adequate design of a micromechanical tuning fork gyroscope. This
would make it possible to determine the performance characteristics of the gyro
including its response, mechanical properties, electrical properties, and noise
levels.
4) A comparison of the open loop performance capabilities between the current
configuration and a tuning fork.
5) A conclusion on whether a tuning fork gyro could be successful on a
micromechanical level even though all previous development efforts failed.
The thesis begins with a description of the gyroscopic properties of a tuning fork and
the micromechanical gyro model to be studied. The equations of motion for the instrument
are then determined rigorously by Hamilton's variational methods. This approach to the
dynamics has not been attempted in the literature and will give greater insight into the system
and the various small order terms neglected in the literature [3,4,5]. The next section of the
thesis analyzes the full equations of motion. Due to the time variant terms in the equations no
closed form solution is possible, and approximation methods are used. First, the
homogeneous equation is transformed into Mathieu's Equation which makes it possible to
approximate the system stability characteristics. The approximated stability region is then
checked by Floquet theory. An approximate infinite series harmonic solution is obtained for
the forced equation of motion to evaluate the various harmonic components as a function of
the system parameters. The approximate system responses are reinforced by a thorough
numerical simulation of the system. The last part of the paper gives design parameters of a
micromechanical tuning fork gyro and calculates a set of perfomance characteristics based on
a thorough mathematical model of the system. The design is done such that the device can be
built using microfabrication techniques, the response is open loop stable with maximized
sensitivity, and the identified noise levels are kept to a minimum. The thesis ends by drawing
conclusions on the feasibility of a micromechanical tuning fork gyroscope as an angular rate
sensor.
1
GYROSCOPE DESCRIPTION AND SYSTEM MODEL
1.1 Tuning Fork Gyroscopic Effect
The operating principles of the tuning fork gyroscope are straight forward. It is
possible to explain the gyroscopic torque of the device using Newton's laws, a freshman
physics background, and referring to the generic schematic of a tuning fork given in Figure
1.1. This description of the gyroscope is not adequate for analytical purposes, but it should
give the reader a clear understanding of the operating principles.
The two fork tines are forced to vibrate 180 degrees out of phase with each other at
the same frequency, od. The vibration causes the structure to have an oscillating mass
moment of inertia about the y-axis, which is the fork axis of symmetry and will hereafter be
labeled the sensitive axis. The principal inertia about this axis can then be written as the sum
of a constant term and an oscillating term,
*
Iy = I0 + Ilsin cot.
(1.1-1)
In addition to the forced tine vibration, assume that the tuning fork has only one degree of
freedom which is rotation about the torsional fork stem. The stem acts like a linear torsional
*
spring with stiffness kt. Further assume that the system is damped linearly, and let the
damping coefficient be labeled bt. The spring and damping act as restraining torques such
that
STy
= - kt - bt.
(1.1-2)
Let the base of the fork be fixed to a specific object which is being rotated with respect to an
inertial reference frame, such that the component of the rotation vector along the fork
sensitive axis is denoted by Q. From Newton's laws the sum of the torques about the
sensitive axis equals the time rate of change of the angular momentum component such that
ITy
-kt0 - bt
dt (Hy
*
=
d
dt[( I0 + Il1sin dt ) (0 + )]
(1.1-3)
Differentiating the right hand side and neglecting the 0 terms, the fork equation of motion
becomes
(1+
sin odt) O+(.L +l-+
cosodt ) 6 +n2
=
cos dt
(1.1-4)
where Con is the torsional natural frequency of the fork,
2
kt
n,
O
Y
Sensitive
SAxis
Forced
Vibration
k tines
fork heel
(rigid)
torsional
stem (weal
in torsion,
in bending
base
(fixed to
platform)
Figure 1.1
Tuning Fork Schematic
A rigorous derivation that includes an analysis of the neglected terms of Equation (1.1-4)
is the basis of Chapter 2. However, this simple derivation gives insight into the physics of
the tuning fork gyroscope. For instance, it is clear that the gyroscopic torque term,
comes from the oscillating angular momentum about the sensitive axis and is proportional to
the magnitude of the angular rate component about the sensitive axis. It is also evident that
the gyroscopic torque is oscillatory at the same frequency as the vibrating tines yet in
quadrature with it. It appears that an instrument can be designed that transforms the
oscillating gyroscopic torque into an output signal. This is accomplished by measuring the
magnitude of an oscillating angular displacement of a torsional spring. The measurement
signal can then demodulated to determine the applied rate direction, and filtered to eliminate
sensitivity to lower frequency torques.
Obviously, the tuning fork instrument has little resemblence to the conventional
rotated wheel gyroscope. Conventional gyroscopes are best characterized by an angular
momentum vector produced by a spinning member which tends to remain fixed in space.
Conventional gyroscopes detect angular rate about a particular axis by measuring the
displacement of the spinning axis relative to a gimbal axis. The magnitude of the
displacement is proportional to the applied rate about a sensitive axis. On the other hand, the
tuning fork instrument has an oscillating angular momentum vector, and yields an AC
signal, which has its magnitude at a specified driving frequency proportional to the applied
rate about a sensitive axis. Previous papers avoided the label gyroscope in describing the
tuning fork angular rate sensor. This was meant to keep the reader from thinking of the
tuning fork gyroscopic reaction as produced by some fixed angular momentum vector. In
this paper, however, the label gyroscope is used to describe the micromechanical tuning
fork instrument. The reason is simple. The famous French physicist Jean Bernard Foucalt
coined the term from its Latin translation to view rotation. The prefix gyro thus stems from
the sensed angular rate, and not the instrument's rotating member.
Throughout the years, many people have had difficulty obtaining a physical feel for
the action of a conventional gyroscope. With the exception of the spinning top toy, objects
with a spinning member are generally not free to be rotated. Therefore, the gyroscopic
reaction is not often observed in nature. However, there are several cases in nature where an
oscillating body is rotated. For one example, consider a child playing on a swing set. The
child, standing on the swing, is set in motion by a friend pushing him, which is the same as
an external torque applied to a rotational system. Slowly, but surely, the swing motion will
damp out and the child's ride will end. But, as discovered by many ten year old
experimentalists, if the child begins to squat up and down on the swing at the proper
frequency, the swing will remain in motion. The squatting produces a periodic variation in
the system mass moment of inertia, and hence causes an oscillating torque if it is in phase
with the swing oscillation. This torque is proportional to the swing's angular velocity which
is analogous to the tuning fork gyroscopic torque.
1.2 Demonstration Model
Even though it is likely that many people, at one time or another, have squatted up
and down to force a swing, the reader may remain unconvinced that a tuning fork can sense
angular rate. This leads to the first goal in the development of a micromechanical tuning fork
gyroscope: to physically demonstrate that a tuning fork can produce a measurable
displacement which is proportional to an applied angular rate. To show this, a large scale
model was designed to serve the function of a classroomdemonstrationmodel . It is
intended to give greater insight into the physical properties of the device.
The model is shown with dimensions in Figure 1.2. The structure may not look
much like a tuning fork, but it has the same basic characteristics. The model consists of a
structure closed at both ends containing only a single tine. A closed fork is preferred to an
open-ended structure to provide structural integrity and symmetry. For practical purposes a
single tine is not desirable since it produces an inertia function which is unsymmetric about
the sensitive axis. Unsymmetry leads to unbalance torques and cross-axis sensitivity.
However, the demonstration model is operated by simply exciting the tines at their
fundamental frequency. A double tine structure is not feasible since it would require an
identical tuning of both tines.
For the single tine the inertia function becomes
I
where
= (I + I)
2
Ii1= Imax - 10 , and
co
+ -L sin 2ot t
2
(1.1-5)
= driving frequency (tine fundamental frequency).
This differs from the double tine symmetric inertia function given in Eqn. (1.1-1) for two
tines. For the single tine, the gyroscopic torque will be vibratory at twice the driving
frequency and proportional to half of the oscillating inertia component.
The model was fabricated using beryllium copper sheets, aluminum, bronze, and
wood. The single copper tine is fixed at both ends by a sandwich joint between aluminum
blocks. Attached at the midpoint of the tine is a bronze block which acts as the high density
inertial proof mass. There are two torsional flexures at each end of the device, along the
sensitive axis, made from thin rectangular copper tabs which are sandwiched into the
aluminum wings and secured to the frame. The frame is attached such that the tension in the
tine, and the length of the flexures, can be easily altered. This flexibility is necessary to
accurately tune the device since for any appreciable gyroscopic torsional vibration the model
must be operated at resonance. This means that the torsional resonant frequency of the
flexures must be twice the fundamental frequency of the tine as seen from Eqn. (1.1-5).
The model was designed such that the tine and flexures were stiff enough to hold
sufficient energy to reduce damping, yet flexible enough to yield a visible frequency. The
flexure torsional stiffness was assumed to follow the equation,
G 3 16
Kt = r ab [3-
3 .3 6 ba+
b5 ]
0. 2 8 (-)
where,
G = torsional modulus of elasticity
L = flexure length
1
a = 7 flexure width
(1.1-6)
1
b = 2flexure thickness
which is given by Roark and Young [7]. The fundamental frequency of the tine was
determined by assuming a fixed-fixed beam, and using Dunkerley's equation [8] to
accomodate the effect of the inertial mass, so that,
2 =
112 t222
Ol l2
where,
)b = W
1 1 2+(02 2 2
+0)222
011 2 = 24.7mbL3)
and
(0222= 6(mL7)
(1.1-7)
o b = tine fundamental frequency
l 1 = fundamental frequency of beam by itself
(022= fundamental frequency of proof mass on a massless beam\
mb = mass of tine
mm = mass of proof mass
L
= tine length
Dunkerley's equation is a useful technique in analyzing structures that have known
stiffness properties, yet have significant masses attached to them. The equation gives a
lower bound for the beam frequency by solving the characteristic equation of the system
flexibility matrix, and neglecting the higher mode components.
For the model, the frequencies were calculated at con = 56.3 Hz and cob = 20.1 Hz.
Tuning the device such that on = 2cb was accomplished by applying tension to the tine to
push up the tine frequency while lengthening the flexures to decrease the torsional
frequency, as indicated in Figure 1.3.
ntr
ITI
V#8 Saw
h""Lv"
rulI**W& I
PQres
K~
5*4amCa
VI EW
TOP
-..
.
. ....
.
ID
"lilb•
1 0/
,•-4
t -
. pins
[
yS
CWS
t5t
rtPWh
'LEWItet To wolb
AV
ldit
SI bE VI EW
Figure 1.2 Demonstration Model
11
twj
5A
N
pae(
Flexure natural frequency vs
Tine natural frequency vs
flexure length
applied tensile force
I
22
L
slenth (m)
Figure 1.3
P teb kfae
Resonance Tuning of Demonstration Model
Several observations were made on the model by vibrating the tine and applying an
angular rate, and they are listed below:
1. The forced vibration excites a twisting mode of the tine about the sensitive
axis. This mode must be of a sufficiently higher frequency than the driving
frequency since it results in an error vibration of the sensing wings.
2. The manually excited tine vibration damps out in less than five seconds due
mostly to energy losses in the joint at the wing, and vibration wave
reflections back into the beam. This indicates that the device design should
have no joints and a stable node where the tines are fixed to the wing.
3. Any misalignment in the z-direction of the proof masses results in an
unbalance torque about the sensitive axis. This should be minimized, yet
does not provide an error since this torque is in quadrature with the
gyroscope torque as shown later.
4. The thin torsional flexures, with sufficient tension applied, are very stiff in
all degrees of freedom as desired.
5. The higher frequency torsional vibrations damp out slowly, mostly due to
viscous air flow.
6. Tuning the system was difficult to achieve and maintain. This indicates that
the tines should be driven off resonance if possible, and a feedback loop to
lock the driving frequency to the torsional resonance is desirable.
7. Gyroscopic action appeared to occur, but was difficult to verify due to tine
damping and unbalance vibrations.
1.3
Micromechanical Configuration
Although the model failed to accurately verify the gyroscopic action, it did indicate
that a gyroscopic torque can be produced. The model also provided a set of useful
guidelines in proceeding with a micromechanical design. The most obvious requirement is
that the device should have only two degrees of freedom: a forced lateral tine vibration and
rotation about the torsional flexures. A simple, yet practical configuration is a double tined
fork, closed at both ends, with the entire structure homogeneous and etched from a silicon
chip. Attached at the midpoint of the tines is a high density metal, which serves to increase
the oscillating inertia. The flexures are attached to the substrate at each end of the sensitive
axis, and the entire fork is suspended over a well which allows the torsional vibration. The
configuration is shown in Figure 1.5. The geometry of the configuration is simple enough
that it is safe to assume that it can be fabricated by current micromachining processes such
as photolithography, masking, doping, selective etching, and electroplating or metallization.
The tines will be driven electrostatically by applying a sinusiodal voltage between
them. The output signal will be produced by detecting the change in capacitance between the
two fork wings and electrodes either suspended above the fork , or buried in the substrate
beneath it. This technique has several advantages. By keeping the driving force within the
fork, coupling betweeen motor and sensor is eliminated by assuring that no motor force is
transfered across the sensor gap. This coupling might represent an error source in the
micromechanical gyroscope designed by Boxenhorn [1,2], and in the tuning fork gyroscope
designed by Sperry [3,6]. Also, the properties of the silicon allow voltages to be applied on
the tines without laying wires across the structure. This configuration also reduces damping
since there are no joints, and the only air damping results from the sensor plates on the
wings. Various other methods for producing a micromechanical motor have been studied in
the literature [9] but electrostatics has been the most successful. The test results of the
Boxenhom micromechanical gyroscope [1] demonstrate the validity of this method.
This configuration appears to offer the benefit of allowing a significant magnitude of
11
the ratio !, which is the proportional to the gyroscopic torque, while maintaining only two
degrees of freedom. The subsequent chapters will provide a thorough analysis, design, and
performance evaluation, based on this configuration.
VO + VI slnwt
substrate
rsional flexure
nsor wing
vibrating tine
high density mass
VO
+ V1
sinwt
SI
sensing electrodes
,7~7
/
/
substrate
//
/
/
7/
/
/
/
Figure 1.5 Micromechanical Configuration
i
/
2
DYNAMICS
2.1 System Model
The device is modeled as a dynamic system consisting of only one degree of freedom.
Figure 2.1 shows a simplified representation of the system. It contains a rigid oscillating
inertia, which represents the fork structure, attached to a torsional spring. The spring is
rigidly attached to a base. It might seem ambiguous that tine vibration is not considered a
degree of freedom, but since its motion is constrained by the motor and its dynamics are
neglected, the tine vibration is accounted in the fixed oscillating inertia term. The tine
dynamics are neglected by assuming that the driving frequency is significantly lower than the
tine fundamental frequency. Also, the motor is isolated sufficiently such that it exerts no
torques on the system. The only external torques acting on the system are the linear spring
torque, kt0, and a viscous air damping torque, btO. The geometry is assumed to be
symmetric such that any gravitational field will not result in any moment about the sensitive
axis.
Three separate coordinate axes are needed to describe the system and are shown in
Figure 2.1. The x-y-z coordinate system is fixed to the base of the fork which is assumed to
be strapped to a moving body. The x'-y'-z' axes are fixed to the fork. The y' axis
corresponds with the y axis, and is the sensitive axis. The x' axis runs through the center of
the fork tines, and is the axis of the lateral tine vibration. The fork-based x' axis is related to
the base-fixed x axis by the rotation angle, 0. The z and z' axis are defined by the right hand
rule such that the z axis is normal to the substrate plane and the z' axis is normal to the fork
plane. The third set of axes represents an inertial reference frame and is denoted by X-Y-Z.
Define the angular rotation of the fork base with respect to the inertial frame as the
vector
16
Abi
=
Wx ix + fl y + Wzi
z
.
(2.1-1)
The component about the y axis is denoted by 0 since it is the angular rate that the
gyroscope is designed to detect. The angular rotation vector of the fork with respect to the
fork base is,
afb = 0ly.
Then, the angular rotation of the fork with respect to the inertial frame, referenced by the x-y-
z coordinates is,
fi= fb + Wbi
=
Wx ix + (a + 0) iy + Wz iz.
(2.1-2)
The inertia of the fork is defined as a tensor referenced by the x'-y'-z' axes. These
axes are chosen because they represent the principal axes for a symmetric design. Then, the
inertia tensor can be written as
I =
Ix00
0 Iy(t) 0
[
0 I(t)J
(2.1-3)
The principal inertia components about the y' and z' axes must be written as a function of
time due to the lateral vibration,
Iy (t) = 10 + Ilsin Odt , and Iz (t) = Iz + AIz sinodt,
(2.1-4)
where Od designates the motor driving frequency. The fork axes and the fork base axes can
be simply related by the rotation angle 0,
ix = ix' cosO + iz'sinO
iy= iyl
i z = iz' cosO - ix' sine.
(2.1-5)
Substituting (2.1-5) into (2.1-2) gives the angular rotation vector in terms of the fork axes,
..i = (Wxcos0 - WzsinO) ix' + (Q + 0) iy, + (Wzcose + QsinO) iz (2.1-6)
This represents the angular rotation relative to the the fork principal axes that the device inertia
is subjected to.
17
y, y'
sensitive axis
oscillating inertia tensor
lateral
vibration axis
air damping torque
le
base
inertial reference frame
Figure 2.1 System Model
2.2 Neglected Modes
The single degree of freedom model of Figure 2.1 is valid only if the entire fork is
rigid and free from structural vibrations. However, as observed from the demonstration
model, the high frequency motor and structural unbalances excite other vibration modes.
Three modes that demand concern are tine twisting, flexure bending about the x'-axis, and
sensor wing bending about the sensitive axis.
The concept of rigidity is often loosely defined. In this study it will be assumed that
if the free vibration frequency for a specific structural mode is significantly higher than the
motor frequency, the mode will remain unperturbed. Significantly higher is generally meant
meant to mean more than a factoroffour larger so that the dynamic load factor [7] from the
motor is approximately unity.
It is now the intention to show that for preliminary micromechanical design
dimensions the three aforementioned modes can be neglected. The modes are analyzed
individually as follows:
Tine Twisting: The tine is modeled as a vertical beam fixed at both ends, constrained from
warping, and excited by an unbalance moment, T. The moment is applied about the tine
longitudinal axis which is parallel to the sensitive axis. Figure 2.2 shows the model, and the
torsional spring constant for the tine is given by Roark and Young [7] as
k
=
0.3 G h w 3
Lt
.
(2.2-1)
The geometric dimensions are given in the figure. The free vibration frequency is then ,
0ob2 =
•
19
where It is the inertia of a single tine and the attached proof mass about its longitudinal axis.
The preliminary design dimensions yield the approximation,
1
It = 1o.
The driving frequency is assumed to be set at the torsional resonance of the flexures which
can be determined from the stiffness Eqn. (1.1-6). Substituting, and comparing the twisting
frequency to four times the driving frequency,
0.9 G hw
Lt o
3
> 16 5.3 Ga b3
L10
Io
(2.2-2)
which is valid provided,
h w3
Lt
80 a b3
Lf(2.2-3)
A suitable design, where the tine twisting is negligable, is when w > 5 b.
Flexure Bending : Each flexure is modeled as a beam fixed at the substrate end and subject
to an excitation moment about the wing end. Assuming a Bernoulli - Euler model which
neglects shear deformation and rotational inertia [9], the free vibration equation is
4
Apdu(y, t) + EI
u(y,
y4
= 0
where
A = cross sectional area of the beam
I = beam inertia about its neutral axis
p
=
density of the beam material
u (y,t ) = deflection curve of the beam
20
(2.2-4)
A solution of the form u ( y, t) = u ( y ) ei
t
is assumed, and fixed end - free end
boundary conditions are applied. The lowest eigenvalue solution to the transcendental
vibration equation [9] is,
0o2 = 12.4 E If
f4 p Af
(2.2-5)
where the subscripted terms denot the beam inertia, length and cross sectional are of the
flexure. Make the approximation E 2G, substitute If = ~ (2a) (2b) 3 , and compare to four
times the driving frequency as given by (2.2-2) yields,
1
mfLf2
40.4
-.10
m-f2
(2.2-6)
This statement is generally satisfied since the mass of the flexures are an insignificant portion
of the structure mass.
Wing Bending:
Each sensor wing is assumed to be fixed down its centerline, along the
sensitive axis, at its attachment to the flexure. The angular rotation, or the tine vibration,
could excite a bending moment about the y-axis, as shown in Figure 2.2. Again, using a
Bernoulli - Euler beam model with fixed end - free end boundary conditions, the
fundamental frequency is given by Eqn. (2.2-5). To correspond with the notation used for
the flexure bending, let the wing dimensions be denoted by
Lw
=
distance from sensitive axis to end of wing
aw
=
one-half of wing width
bw
=
one-half of wing thickness
mw
=
mass of the wing.
Comparing the fundamental wing bending frequency to four times the driving frequency
results in,
21
2.1 aw bw 2
Lw 3 mw
40.4 a b3
Lf 10
To approximate, assume the dimensions are such that
mwL
=- Io and
3
Lw = 5Lf.
Then, the constraint becomes
aw b3 2 20 a b3
(2.2-8)
which thickness
isachievable forconsiderably
awing
greater than the flexure thickness(2.2-8)
which is achievable for a wing thickness considerably greater than the flexure thickness.
22
SZ
u(y,t)
O4
L
7
y
Flexure Bending
T
A
!
•
_Jl
-e
i
m, G
t
|
z
Tine Twisting
f%
"
~
v
,%rhA
u(xt)
z
Wing Bending
Figure 2.2 Neglected Modes
2.3
Equations of Motion
A simple, direct approach to the system dynamics was undertaken in chapter 1
resulting in the equation of motion given by Eqn. (1.1-4). In the process, several
approximations were made and the analysis was only one-dimensional. In order to account
for the cross-axis terms, Hamilton's variational approach to the system dynamics is
performed which gives a more rigorous, indirect derivation of the equations of motion.
Hamilton's method uses an energy technique to determine the equations that the
system variables, or state vector, must obey so that the system remains at equilibrium with
the external forces and constraints. Hamilton derived a variationalindicator which measures
the magnitude of deviation from equilibrium due to incremental changes in the state variables.
The variational indicator is represented as an integral over the trajectory time period for a sum
of system work increments due to the variable deviations. The equations of motion are
obtained by setting the variational indicator to zero for arbitrary admissable state variations.
Joseph-Louis Lagrange derived a general equation [10] for a system that obeys Hamilton's
principle, and which bears his name,
~~T
T
-av
a4 j
for j = 1...n
where,
j
(2.3-1)
n = the number of state variables, and
T = system kinetic coenergy
4j = generalized coordinate for each state variable
V = system potential energy
'j = sum of all nonservative forces in the 4j dimension.
The reader is referred to Crandall et al [10] for a complete description of Hamilton's
variational methods.
24
From the single degree of freedom system model illustrated in Figure 2.1, the only
generalized coordinate is 41 = 0. The only nonconservative force acting on the system is the
linear damping, so that Zl = btL.
The kinetic coenergy is due to the angular rotation of the
fork ,
T =
1
.fi
O IL fi.
(2.3-2)
It should be noted that the system kinetec coenergy is equal to the kinetic energy since the
system is defined in a Newtonian reference frame. Substituting for the inertia tensor Eqn.
(2.1-3) and the angular rotation vector of Eqn. (2.1-6) gives,
T = •
+,(xcos-Wsin)
I Y(t) (Q+ 2 +1 Iz (t) (Wcose+Wxsine)
x
2
The torsional spring restraining force is the only source of potential energy, so that,
V =
kt 0 2
2
For the system, Lagrange's Equation (2.3-1) takes the form,
d DT I_--aTu-+ aV
---
= •0
where,
dT= --lyI
9dt aeT
-
I)and,
(t +
= ly(t)
+ +l)
_T= -Ix Wxcos- Wzsin0Wzcos+Wxsin
and,
+ Izt •WFos+W xsin
Wxcos_-W sine)
(2.3-4)
Putting it all together, Lagrange's Equation becomes,
25
Iy(t) (6+0+ ly(t) (+0)
+Ix( WxcosO-Wzsin0X Wxsin0+Wzcose) -
Iz(t) (WzcosO+WsinOX w.cosO-wsino) +k19 +bto =0.
(2.3-5)
The trigonometric terms in (2.3-5) can be expanded in a Taylor series to transform the
equation of motion into algebraic form. To simplify the expansion it can be noted that the
rotation angle, 0, is very small. For example, the rotation angle is on the order of 10-5
radians for the Boxenhorn gyroscope [1,2], so that first order approximations will result in
negligable errors. Letting sine0
0, cos0 = 1, and sin 2 0 = 0, and also substituting (2.1-4)
for the oscillating inertia functions, the equation of motion becomes
(lo + isindit+)
IodcOSd+e)
1+
+ (wx-
I - Iz(t))0 +'kt +bte = WxWz(Ix - Iz(t))
(2.3-6)
This second order differential equation represents the complete description of the
tuning fork gyroscope subject to an angular rate vector. In the literature on tuning fork
gyroscopes, only Fearnside and Briggs [5] included the cross-axis and small order terms in
the dynamics analysis. However, the analysis done by Fearnside and Briggs differs from this
paper in a few respects. Fearnside and Briggs assume the damping is of large magnitude,
which is not the case on a micromechanical level. They are then able to neglect second order
terms of the ratio Il Od . This reduces the significance of the time-varying terms present for
a lightly damped system which is studied in chapter 3. They also use a direct approach to the
dynamics and simply drop all cross-axis terms after noting that they limit the device accuracy.
The first simplification made on Eqn. (2.3-6) is to neglect the angular acceleration
term, Q.
The driving frequency is assumed to be high enough such that the frequency of
the input rate is negligable in comparison. It is then possible to drop the 0 term from the
equation of motion. It must be noted that a limitation is placed on the system bandwidth, but
the limitation is insignificant for practical applications. The driving frequency for a
26
equation of motion. It must be noted that a limitation is placed on the system bandwidth, but
the limitation is insignificant for practical applications. The driving frequency for a
micromechanical device is assumed to be in the 1-5 kHz range, and applications in guidance
or control areas do not require bandwidths greater than 100 Hz. The input frequency is thus
negligable compared to the output frequency, and no significant error results from dropping
the f term. The equation of motion can now be written in a more convenient form,
(1o + I1 sin odt) 0 -{ bt + I1 Od COSOdt) 0 + kt0 = -I1Z Od COS odt
2 x- W•) 0] sin Codt
+(Ix- I~o WWz - (W2w
- ) 0 - III WxWz +(W
(2.3-7)
2.4 Simplified Output Solution
A closed form solution to Eqn.(2.3-7) is not apparent. The first step to understanding
the behavior of the system is to drop all terms that complicate the equation and can be
reasoned to have minimal effect on the solution. The cross-axis terms on the right hand side
of the equation will be dropped first. These terms comprise a torque about the sensitive axis
that is much smaller than the spring restraining torque, the damping torque, and the
gyroscopic torque. The following section will give a more thorough analysis of the crossaxis effects. The equation can now be transformed to a solvable form if I1is considered
much smaller than 10, and Ilw is much smaller than bt. The equation is then a second order,
constant coefficient, harmonically forced oscillator,
Io0 0 + b+k
= -I10
cd COS Od t.
Assume a solution of the form,
0 = A sin odt + B cos odt,
substituting gives,
27
(2.4-1)
A=
-I
0,2 bt
-Io(kt - lo a) + kt ( kt - lo w ) + w2 b2
and,
B = kt- Io o
.
(2.4-2)
The rotation angle can be maximized if the driving frequency is set at the torsional natural
frequency such that,
Md
o =
(2.4-3)
Hereafter in this paper, when the frequency is denoted by the symbol o without a subscript,
it denotes the driving frequency tuned to the torsional resonant frequency. The output is then
a pure sine curve, in phase with the motor vibration,
8 (t)
=
bt
sin ot .
(2.4-4)
The linear damping coefficient can be written in terms of the system quality factor, Q ,
where
bt =
I00
-p
. A more convenient expression for the simplified output solution is
0 (t) =
-IQQ
I0(
sin aot.
(2.4-5)
If all the assumptions for the simplified solution are valid, the output angle is
proportional to the ratio of the oscillating inertia component to the nominal inertia component,
the system quality factor, the magnitude of the applied rate, and inversely proportional to the
driving frequency. This gives a calibrated measure of the angular rate about the sensitive
axis. The output signal can also be demodulated with the driving oscillation as a reference
28
2.5 Simplified Output Transient Response
Eqn. (2.4-5) gives the steady state vibration of the harmonically forced, linear
system. However, the second order dynamics of the tuning fork will lead to an oscillating
transient period that must be investigated in terms of system bandwidth. Eqn. (2.4-1) can be
written in the following form,
**
-I
2
*
-
n e
-+ "+
Q +
COd
COS Od t .
--
n
0
(2.5-1)
Let II (t) = I1 sin Odt, then Eqn. (2.5-1) can be written,
0
2
*•
Q0 +
-i (t)
10
o
Since this equation is linear, Laplace transforming yields,
-
-
S
I1
0
s
- G (s) .
s2 + ýS +Cn2
Q
(2.5-2)
G(s) represents the system transfer function, but it can not identify the system bandwidth
since it relates the input signal which has unknown frequency to an output signal oscillating at
the driving frequency. In order to transform G(s) into a DC transfer function, Gm(s), the
output signal is frequency modulated at the driving frequency. This means,
0m (t) = 8 (t) e1
Om (s)
=
or, in the Laplace domain,
Od
(s +j COd)
Replace s by s + j wd to give the DC transfer function,
-I
s +
---
m
j Cd )
o
(s + j Od)2 + -(S
Q
29
=
+ j (0d)+5
Q
Gm (s) .
(2.5-3)
Expanding, rationalizing, and setting the driving frequency at the resonant frequency,
Gm (s)
s +j0)[s s+ 2jws+
+S2 + _.Q2
4(os + f2(S
4c 2s
+2•
1Q+
(2.5-4)
Assuming that the system bandwith is much lower than the driving frequency , the
components of the transfer function can be written,
(s
I{s +
Imag (Gm (s)) =
4oýs + Q)2
and ,
- Ii
Real.(Gm (s)) =
I
21s +2(Q
(2.5-5)
It is clear that the quadrature component is small compared to the in-phase component so that
the signal has negligable phase shift during the transient period, and the sytem bandwidth is
BW = - .
2Q
(2.5-6)
2.6 Cross Axis Terms
The right hand side of Eqn. (2.3-7) consists of the gyroscopic torque term plus four
terms due to the input rate about the axes orthogonal to the sensitive axis. These terms lead to
torques about the sensitive axis of unknown magnitude. But, if these torques can be shown
to either have negligable effect on the system, or cause an output deflection that has no
component that is in-phase with the motor at the driving frequency, no error signal will
result. This is because appropriate signal filtering and demodulation can eliminate all signals
from the sensor rotation except the vibrational component in phase with the motor within a
30
~
component that is in-phase with the motor at the driving frequency, no error signal will
result. This is because appropriate signal filtering and demodulation can eliminate all signals
from the sensor rotation except the vibrational component in phase with the motor within a
small frequency band about wd. However, any quadrature or off-frequency signal must not
exceed the designed output range of the gyroscope.
The cross axis terms are analyzed term by term:
1) (Ix - IOz)(WxWz) : This causes a DC output which can be filtered out. The bias that it
causes can be made as small as possible by designing the nominal inertias about the x and z
axes to correspond. It appears from Figure 1.5 that the symmetry can be easily obtained by
adjusting the x and y dimensions of the proof masses.
2) (Ix - Iz)(Wx 2 - Wz 2) 0:
For a symmetric design where Ix= I0z, this term can be
regarded as a small deviation in the spring constant. But, assuming a driving frequency in
the 2-4 kHz range, and a practical limit on the cross axis rates of around 20 Hz, then kt = 6 2
10 >> (Ix - I0z)(Wx 2 - Wz2 ). Hence, this term will have less than a 0.01% deviation on
the spring constant and will not alter the output significantly.
3) AlzWxWz sin or : AIz is the oscillation inertia about the z-axis, and is thus small and
on the order of Ii. Again, for a high frequency system, this term is tiny compared to the
gyroscopic torque for any significant Q. It is also in quadrature with the gyroscopic torque,
so even for very small values of 9, the majority of the error signal from this term will be
eliminated by demodulation.
4) Alz(Wx 2 - Wz2) 0 sin ox : This term can be viewed as a high frequency deviation in the
spring constant. Since Ilz is generally on the order of 1%of Ioz, the maximum deiviation in
the spring constant from this term is only about 0.0001%. And the frequency component of
31
the term does not lead to difficulties since, when combined with the sinusoidal output, it
results in a double frequency torque.
In general, the cross axis terms do not lead to a significant limit on the accuracy of
the device. As shown in the subsequent chapters, for a micromechanical device, Brownian
and electronics noise are the predominant error sources. Zero rate errors, such as cross-axis
coupling, tend to be insignificant by comparison.
32
3
SYSTEM RESPONSE ANALYSIS
In this chapter, the system response to the complete equation of motion will be
studied. For the tuning fork to operate efficiently as an angular rate sensor, the output angle
must be a linear function of the input rate. Eqn. (2.4-5) shows a linear relationship, but in its
derivation the periodic terms in the equation of motion were neglected. It is necessary to
verify that the periodic terms have a negligable effect on the system response. A brief
overview of linear periodic equations is presented, and then the tuning fork equation of
motion is analyzed by several different analytical and numerical tools.
3.1 Periodically Time - Varying System
If the cross axis terms are neglected and the system is driven at its torsional resonance
so that on = COd = 0o, the complete equation of motion (2.3-7) becomes,
\
1 sin)t
+
+
.Io
Io
(1 + esin ox 0 +
cos
+ 02 8
-
1
0 Cos ct
or,
Io
+e )cos 0x
+ 0 0 =-
Q )cos 0t .
where
e =
Q =
the mechanicalgain
sytem quality factor.
Generally, the mechanical gain is a small number, and the quality factor is defined in
Eqn. (2.4-5). Equation (3.1-1) is a linear, time varying, periodic differential equation (LP)
of second order. Equations of this form have arisen in many different fields of application,
33
and hence many different analytical methods to describe the behavior of such systems have
been studied. The problem is that there is no uniform approach, such as the Laplace
transform for linear, time-invariant, differential equations, to describe these systems. There
are also no known exact solutions, which makes it necessary to treat each equation
individually by the most appropriate approximate method.
The equation of motion (3.1-1) is of the form
a (t) x + b(t)x + c(t) x = d(t)
where a, b, and the forcing function d, are all periodic of period T = 2 x / co . The main
deficiency with the simplified solution of Chapter 2 is that even though a(t) = 1 since E is
small, the periodic component of b(t) cannot be accurately neglected since 1 / Q is on the
same order as E for a lightly damped micromechanical system. This type of equation, with
nearly constant a and c and periodic b , is a rare and unexplored form of an LP equation.
Intuitively, the response to Eqn. (3.1-1) will be approximately the steady state and transient
response of sections 2.4 and 2.5 for a small value of the product parameter, EQ. But when
EQ grows greater than unity, and the periodic component of b(t) begins to dominate, the
oscillating response will deviate from the simplified solution. This can be viewed as a
periodic perturbation in the damping, so that as EQ increases the output frequency should
remain constant, but the output magnitude, phase, and bandwidth will be affected.
The periodic variation in the damping is an effective energy pumping of the system.
This results in a substantial increase or decrease in the amplitude of the output oscillations. A
physical example ofenergy pumping from the circuit design field is presented by Richards
[11]. By periodically changing the capacitor plate distance in an LC circuit network, it is
possible to pump the voltage across the capacitor, which results in a periodic damping term.
When done at the proper frequency and phase, this increases the magnitude of the voltage
across the capacitor. This view leads to the conclusion that for specific values of the
parameter eQ the output magnitude will increase without bound, and the system will cease to
34
be globally stable. The product of the mechanical gain and the system quality factor, EQ, will
thus be referred to as the stabilityparameter.
The stability problem of periodic systems is unique because small variations in a
periodic coefficient can cause the system to lie in a region of instability, separated by regions
of stability. In the tuning fork case, this means that as eQ is increased past a certain limit the
response could go unstable, but a continued increase in the parameter could stabilize the
system. The stability problem of periodic systems has been studied in depth, with particular
emphasize by Maclachlan [12], Richards [11], and Porter [13].
An optimized design of the micromechanical tuning fork gyroscope will maximize the
parameter eQ, since it is a factor in the output magnitude as shown by (2.4-5). It is thus
imperative to determine the stability characteristics of this parameter. Several techniques to
determine the stability of periodic systems exist. They include transformation to the Mathieu
equation, Lyapunov stability theory, Floquet theory, and numerical simulation.
In the literature, much research has been done on homogeneous equations of the
form
K +[a- 2q2
(t)]x = 0 ,
where V (t) is a periodic function. This is known as Hill's equation and has its roots date
back to 1886. A special case of Hill's equation that has received particular attention in regard
to stability is the Mathieu equation, and is given by
x + [a - 2qcos2t]x = 0 .
(3.1-2)
Approximate periodic solutions to Eqn. (3.1-2) have been obtained as a function of the
parameters a, q. Since the stability regions for the parameters have been well documented
[11,12], a useful technique to determine the stability of a particular LP system is to transform
it to the Mathieu equation. Then the characteristics of the transformed equation, plus the
particular transformation, yield information about the original equation. The disadvantage of
this approach is that an appropriate transformation must be found, and information can be lost
in the transformation.
35
Another approach is based on Floquet theory. This is recently emerging as a useful
technique to give an exact yes or no answer to a particular system stability. However, this
approach is limited in that it requires the computation of a discrete transition matrix, and its
respective eigenvalues, for a given set of parameters. For the majority of LP systems, the
discrete transition matrix cannot be obtained in closed form. Numerical techniques must be
used and the stability conclusions are only obtained for specific values of the parameters.
Lyapunov stability theory, a more theoretical approach, uses mathematical functional
analysis to check if a suitable bound can be determined for the system response. It has been
used successfully for many particular periodic equations, but it requires the formulation of a
suitable Lyapunov function. There is no defined method to determine the Lyapunov
function, and extensive trial and error is often the method employed. Unfortunately, the
Lyapunov function is not easily obtained for the tuning fork equation.
The last possible technique, numerical simulation, has the problems of truncation and
round-off error, and is very sensitive to errors around the stability limits. Also, the high
frequency oscillations in an unstable region can grow slowly over many periods, so extensive
computation time is needed to reach conclusions. Simulation, though, can be used to
reinforce analytical results and verify the forced system response.
3.2 Stability in Terms of the Mathieu Equation
In order to transform Eqn. (3.1-1) into a form of the Mathieu equation of (3.1-2), it is first
written in a more convenient form. Dividing through by the a(t) term and noting the first
order expansion
1
= 1 - E sin ot + O (E2 )
1 + E sin ot
36
(3.2-1)
gives,
2 1 - esin oca
)0 +
cos ax) 1 - e sin
+
0 +
= -E
co(I - esinct) Cos ct.
Again, neglecting the second order terms in e, gives
O + o+ E coos ot - Qsin wt i
+
2(
- esinwt)O = -eocosxt.
If, for a lightly damped system, the parameter i is on the order of e, then the sine
component of the b(t) term is negligable compared to the cosine component. This gives
0 + (
+e)
cos
+
t
2 (1 - esinot)O =-0 e tocos a.
(3.2-2)
An appropriate change of variables must be found that eliminates the b(t) term from
the homogeneous form of Eqn. (3.2-2). An appropriate transformation [11] is
x = 0exp- -
b(') d)
x=Bnp(2 ",
(3.2-3)
which transforms the equation
into
0 + b(t) 0 + c(t) 0 = 0
x + [c(t)
1.
- .
(t)
-41 b2(t)]xx0=
0.
(3.2-4)
Substituting the values for a(t) and b(t) from Eqn. (3.2-2) yields the transformed equation,
S+
t2 1 4•-) -- F.sin (°t - 1 -
4 Q2
2
cos
2Q
-2t COS2•t
4
x
= 0.
(3.2-5)
The oscillating component of the second term of Eqn. (3.2-5) is approximated by the sine
component since Q >> 1 and e << 1. The transformed equation is then written
37
x)
+
e sin at x = 0.
4Q /
](3.2-6)
To put this in the form of Eqn. (3.1-2) a change is made in the independent variable. Let,
COt
F.
Eqn. (3.2-6) is then written,
S
2
+
4
2e cos 21 x(t) = 0.
-
d-
(3.2-7)
Equation (3.2-7) is in the form of the Mathieu equation with the parameters
1
a=4 - -2 and q = e.
Q
An approximate solution to the Mathieu equation was first introduced by E. L. Ince in
1932. Since then, many mathematicians have studied the equation, and very accurate
approximate series solutions exist. The reader is referred to Mclachlan [121 for a thorough
treatise of the Mathieu equation and its solutions. A simple explanation of the solution
technique, and its application to the transformed tuning fork gyroscope equation of motion, is
presented here.
Since the Mathieu equation is second order and linear, the general solution is
x(') = Ct xl (T)+ C2 x2 (T)
(3.2-8)
provided that the system Wronskian vanishes identically. For a differential equation of
period T, it has been proven by Floquet [11] that the system solution at any time t is related
to the solution at time r + T by a complex constant factor. This says,
x (t+ T) = a x(r).
(3.2-9)
The two independent particular solutions of (3.2-8) are then
xl (r+ T) = al xl (r)+ a2 x2 (T)
x2 (r+ T) = a3 xl(C) + a4 x2 (T).
38
(3.2-10)
where the a's represent arbitrary complex constants. Combining (3.2-10) with (3.2-8) and
(3.2-9) gives,
[(al -a) Cl + a2 C2 ] + [ a3 C1 + (a4 - )C2] = 0.
(3.2-11)
A nontrivial solution to this equation exists only if
a,-o
a3 I
a2
a4 - o
0.
(3.2-12)
The Wronskian of the Mathieu equation is defined as
W
=
Xl
X2
dxj
dr
dx2
dr
It was shown by Porter [13] that for two independent solutions to the Mathieu equation, xl
and x2,
dW
the derivative of the Wronskian, , is identically zero. Then,
dr
W () = W (0),
and W (r +T) = W (r).
It is seen from Eqn. (3.2-10) that
W (C+ T) = (al a4 - a2 a3) W (c).
Therefore,
al a4 - a2 a3 = 1.
Substituting this into the determinant equation (3.2-12), a relationship is obtained
between two of the arbitrary constants and the periodic scaling factor, a
0 2 - (al + a4)
+ 1 = 0.
A root locus for this equation can be drawn where the free parameter, or gain , is
K = al +a4.
39
(3.2-13)
When K = 2 and K = -2, the roots become a = -1, and a = 1, respectively. At this
point, when the roots a cross from within the complex plane unit circle to outside the unit
circle, the system becomes unstable. This is analogous to the stability criteria of a discrete
system with poles in the z-plane. The instability is seen by noting that if the magnitude of a
exceeds unity the system output of Eqn. (3.2-9) will increase each period. The analogy to
discrete systems is another interesting topic altogether, and it enables periodic systems to be
readily analyzed in discrete state space form. This is the basis to Floquet theory which is
studied in a forthcoming section.
From Eqn. (3.2-9) it is seen that for a = 1 the solution is periodic with period T,
and for a = -1 the solution is periodic with period 2T. The Mathieu equation has period n
, so the goal is to adjust the value of the parameters, a and q , until solutions that have period
x or 2x are found. The corresponding values of the parameters represent the stability
boundaries. It is noted from the Mathieu equation (3.1-2) that as q approaches zero the
system reduces to a free oscillator with solution,
x (t) = C3 sin mt + C4 cos mt ,
for m
2
= a.
Assume an infinite series solution, that is periodic in x or 2x, and reduces to the free
oscillator, in the form,
xl (t) = cos mt + q C11 (t). + q2 C12 (t) + ...
x2 (t) = sin mt + q C2 1 (t) + q2 C2 2 (t) + ...
(3.2-14)
Since there are two parameters, the technique is to fix q, and write a as a power series in q,
a = 1 + at q + a2q2
+
...
(3.2-15)
For both the even and odd solutions, substituting Eqns (3.2-14) and (3.2-15) into the
Mathieu equation [12] yields differential equations for the coefficients Cij. It is shown
[12,13] that the resulting differential equations can be solved for an infinite sequence of
discrete values for the parameter a. These values are known as characteristicnumbers and
the corresponding curves from Eqn.(3.2-15) are the characteristiccurves.
40
Mclachlan [12] calculates the characteristic curves to high accuracy for both the even
and odd solutions, and labels them an and bn , respectively. The curves are plotted in the a-q
plane, and the regions bounded by the curves are checked by substitution to determine if they
represent stable or unstable regions. This stability chart is called theStrutt diagram after the
mathematician who first introduced it. A reprint of Mclachlan's diagram is shown in Figure
3.1. In the figure, the shaded regions are the unstable areas, and the unshaded regions are
stable areas.
For the transformed equation of motion (3.2-7) the Mathieu parameters are
1
a= 4
and
q = e.
(3.2-16)
This point lies very near the b2 curve in the a-q plane shown in Figure 3.1. For this point to
lie in a stable region, the parameters are restricted to place the point below the b2 curve.
Mclachlan's expansion for the b2 curve is truncated to give
1
b2
a = 4 --2 q2 + O(q4 ).
Substituting from Eqn. (3.2-16)
1
b2": 4--
4-
1
2 + O(E4).
(3.2-17)
The stability boundary is then,
eQ =
1
(3.2-18)
For the point to lie in the stable region of Figure 3.1, it is seen that,
EQ < /2 .
(3.2-19)
The approximate stability boundaries for the transformed equation are quite accurate.
The remaining problem is to determine the relationship between the stability of the
transformed equation (3.2-7) and the original equation (3.1-1). First, the change in the
independent variable from t to 'T,
from Eqn. (3.2-7), does not affect the stability
characteristics. It is a scaling and shift on the time axis. Stability criteria involves the limit as
time approaches infinity, thus a scaled shift does not affect stability definitions. The
41
Figure 3.1
Stability Regions for the Parameters of the Mathieu Equation
42
dependent variable change is given by Eqn. (3.2-5). The variable of concern, 0, is equal to
the transformed variable, x, multiplied by a scaling function
0 (t) = x (t)k(t)
where,
1 1
k(t) = exp[-,o(t + esinot)] .
A matter of concern is that the scaling function does not contain one-to-one correspondence.
But, the function is bounded by
kub = exp
t
-
where the initial constant is
koexp
ko = exp ()
t
.
For a high frequency, the upper bound of the scaling function is a large number at time t =
0, then it quickly dies to zero. Therefore, at the upper bound, the scaling function magnifies
the transformed variable, x (t), for small values of time. As time increases the scaling
function drives x (t) to zero. If the variable x (t) is shown to be stable, then the
transformation cannot destabilize the response, so that 0 (t) is also stable. The restriction
given by Eqn. (3.2-19) is thus a sufficient, but not necessary, condition for stability of the
equation of motion.
43
3.3
Approximate Forced System Response by Harmonic Balance
In this section, the effects that the periodic terms in the equation of motion have on
theforced response is analyzed. As discussed in section 3.1, it is believed that the existence
of the periodic terms will produce a phase shift from the simplified sinusoidal approximation
of Eqn. (2.4-5). The system can be written in state space form,
i (t) = G(t) x(t) + B(t) Q
where,
x (t) = x
,
[X2
G(t)
=
,2(1
esincot)
.CO .
-O1 - e sin wt)EMCOS
coscot
B0(t) = [ - e ccos ot
(3.3-1)
The variable xl denotes the output angle 0. The input to the system is the magnitude of
the applied rate about the sensitive axis since only the open loop system is studied. It is seen
that both the plant and forcing matrix can be written in standard periodic form
G (t) = Asys [ sin (ot + tsys) ] and B (t) = Ainp [ sin ((ot + inp) ].(3.3-2)
The A and b matrices designate the amplitude and phase of the periodic functions. A
graphical description of the system is shown in Figure 3.2. The steady state output of the
system can then be written in the same form,
x (t) = Aout [ sin (cot + .out) ].
(3.3-3)
It is possible to determine the amplitude and phase of the system output for a given set of
system parameters. This technique is similar to the describingfunction method to analyze
the effect of system nonlinearities on sinusiodal input waveforms. In the case presented here,
the input signal form is known, and the amplitude and phase of the system for given
parametersis also known. It follows that the output amplitude and phase can be determined.
44
X (t)
(Aout,co,
i dt
u
out
Ow
+
integrator
i
G (t)
(A sys,o,
sys
Figure 3.2 Periodically Forced System
The output waveform can be expanded in a Fourier series.
xI(t) = All (sin ot + 11) = ao +
b, cos rwt
an sin not +
n =1
r= 1
(3.3-4)
This representation for the output 0 (t) can then be substituted into the equation of motion in
the form of Eqn. (3.1-1) to determine the Fourier coefficients. This technique to determine
the amplitude and phase of an output signal is known as harmonicbalance. The derivatives
are
0(t) = C anna cos not n=1
0(t) = -
brro sin rcot
r=1
brr 2 0 2 sin rot.
ann202 cos not n=1
r= 1
Substituting,
45
an n2 o02 sin n(ot +
- (1 + e sin wt)
n =1
n=1
+
r
+
Cos t
r= 1
brr0sin nt
ann(cos not -
b r2 o 2
co ao+
r=1
ansin not +1
n=1
bcos rot
cos cot .
=-
r=1
(3.3-5)
To simplify Eqn. (3.3-5) it is necessary to recall the following trigonometric identities:
sin a sin 3 =
cos a cosl
1
cos (a 1
=
sin a cos 3 =
1
) - 'cos( a + p )
1
cos ( a- 3 ) + Icos ( a + o )
1
1
'sin ( a + p ) + 2 sin ( a - 3 )
(3.3-6)
Using these identities, the left hand side of Eqn. (3.4-5) is simplified and expanded
into an infinite summation of even and odd harmonics. By equating the coefficients of the
various harmonics, an infinite string of equations in terms of the system parameters and the
Fourier coefficients is obtained. The first five such equations are given by,
constant:
ao = 0.
sino(0t:
-~-bl + (2e(c2 - eo.)b2 = 0.
cos ot:
Q at - ( 2 e
sin 2ct:
cos 20t:
02
2
-
ec) a2 = -RD
- E 02 bi - 30 2 a2 - ---
2(02
02 al + 2
0.
) b3 = 0.
-
b2 + (
9
a2 - 302 b2 - ( E02-
3EW
) a3 = 0.
(3.3-7)
The equations proceed such that only four terms are present in each equation. A general form
is noted such that for the jth harmonic the two equations are
sin jot:
2
012 bj. 1 - (j2 -1 )02 aj - j
bj +
( + 1)2
( ++ 1)2
= , and
] b
2
2
2
46
(j [ 1)2
2
02 -
(j +
2
j1)2
eo ] aj+i = 0.
(3.3-8)
The exception to the generalization is that for the cos ot equation the forcing term on the right
hand side is the gyroscopic torque, -
e o.
The sequence of equations can be written in terms of infinite matrices,
Cx= f
o0
(02
0
Q
o0
-2(Eo2
Q
0
where,
-CO
2
-
al
e)
bl
0
E[
0o
-3co 2
(3.3-9)
It is not possible to exactly determine the values of the Fourier coefficients. But, as
previously shown, the system is stable for a specified region of the parameters. Any periodic
input applied to the system will result in a stable, periodic output. Then, it is necessary for
the Fourier series to converge and the coefficients, an and bn, tend to zero for large n. The
infinite matrices can then be truncated at a sufficiently large dimension by neglecting
infinitesimal Fourier coefficients. The corresponding finite matrix is
xf = [ al, b2 , a2, ...,bm]T
where m is the number of
significant harmonics in the solution. The resulting finite matrix equation can be solved
where,
Xf = Cr 1 ff.
(3.3-10)
It is desireable to obtain an expression relating the Fourier coefficients to the system
parameters,
0),
e, and Q. This would give system design guidelines necessary to keep the
quadrature and higher harmonic components of the output small. However, the existence of
47
the matrix inversion makes it practically impossible to obtain an analytical expression for the
Fourier coefficients. Hence, numerical solutions must be obtained for the truncated matrisx
equation (3.3-10) for different parameter values to obtain quantitative relationships.
First, a set of preliminary design parameters that are within the stability region is
chosen where,
oo = 6000 rad/sec
Q = 2000 and
eQ = 1.0.
Substituting these values into a 10 X 10 dimensional form of Eqn. (3.3-9), the solutions are
al = - 1.7 E-3
bl = - 5.6E-7
a2 = 1.85 E-10
b2 = - 2.8 E-7
a3 = 3.4 E-11
b3 = 2.9 E-14
a4 = 5.1 E-18
b4 = 5.2 E-15
a5 = - 8.6 E-19
b5 = - 9.4 E-22
It is encouraging to observe that the magnitude of the in-phase component ( al) at the
driving frequency is equal to EQ
-.
This is the magnitude of the pure sine output that was
determined by neglecting the periodic terms as shown in section 2.5. Also, the Fourier series
converges rapidly. All terms greater than the second harmonic appear negligable. It is thus
adequate to approximate the matrix equation (3.3-9) by a 4X4 dimensional matrix equation.
This assumption loses its validity for large values of EQ since system instability could lead to
a diverging Fourier series.
But, assuming that all terms greater than the second harmonic are negligable, values
for al, a2, bl, and b2 can be determined for a wide range of system parameters. Numerical
results are obtained for values of the stability parameter from eQ = 0.3 to eQ = 30, and for
values of the system quality factor ranging from Q = 200 to Q = 1000. Different values for
the system frequency were inputted, but , for a fixed Q and e the Fourier coefficients were
48
invariant to system frequency. The results are plotted in Figure 3.3, and a frequency a = 1
kHz was used throughout.
The figure shows the phase shift of the first harmonic versus the stability parameter.
The phase shift is defined such that a 90 degree shift results in a pure cosine component, and
a zero degree shift is a pure sine component. Results for three different values of the system
quality factor are plotted, with Q = 200 as a minimum and Q = 1000 as a maximum. The
ratio of the second cosine harmonic to the first sine harmonic is also plotted. For all cases
run, the cosine phase of the second harmonic was significantly greater in magnitude than the
sine component. Hence, this plot gives information on the magnitude of higher harmonic
oscillations compared to the oscillations at the system frequency. The third plot shows the
ratio of the first sine harmonic to the estimated value of the output obtained from neglecting
the periodic terms. The estimated value is given by Eqn. (2.4-5).
The numerical results to the Fourier coefficients yield useful relationships between the
system parameters and the actual response. A number of conclusions can be reached and
they are listed as follows:
1)
The Fourier series converges within four terms in the esitmated stability range
EQ < f12.
2)
The only significant higher harmonic term that is excited is the second even
harmonic ( b2 cos 2mt ). However, this term is less than 1% of the first odd
harmonic in the estimated stability range, and only increases to 3% at eQ = 30.
Hence, all higher harmonic terms in the output can be accurately neglected. The
small magnitude of higher harmonic oscillations that does exist is eliminated by
filtering the output signal.
3)
For eQ < 4, the forced system output at the first odd harmonic (sin cot) is exactly
the output estimated form neglecting the periodic terms in the equation of motion.
49
first harmonic phase shift vs.
stability parameter
S-00
1r000
0
5
10
15
20
30
25
stdtty pmntnv(eQ )
ratio of second cosine harmonic
ratio of first sine harmonic to
to first sine harmonic, vs. eQ
time invariant approximation
1L
12
08
06
04
02
stcabriy pmuar (60)
Figure 3.3
0U
1i
i
1
1
1i
0
5
10
15
20
25
stablity pown.etar(Q )
Numerical Results for Harmonic Balance Fourier Coefficients
3
For 4 < eQ < 8, the sine component decreases at a nearly linear rate. Therefore,
for a system designed within the estimated stability range, the in-phase output is
accurately predicted from the linear simplified theory,
al
=
QEQ
-.
4)
The output phase shift increases with the magnitude of the system damping.
5)
Within the estimated stability range, the phase shift is less than 3 degrees for a
-system with minimum Q = 200. For eQ > 5, the phase shift increases fairly
rapidly. This small quadrature component can be easily eliminated by
demodulation.
3.4
Numerical Simulation
The analytical results of the previous three sections have all included some degree of
approximation. To gain increased confidence in the behavior of the periodic system, the
equations of motion are solved numerically. The role of the simulation is to reinforce the
analyticalresults. All too often, simulation is relied upon to provide the bulk of information
on the behavior of a particular dynamic system. For the dynamic system presented in this
paper, the high frequency nature renders it difficult to reach conclusive results from a
simulation alone. Very small step-sizes and a very large number of steps are needed to solve
the equations for a significant period of time. This requires excessive computation and
magnifies the effect of truncation and round-off errors.
But, the simulation is useful in a few regards. The simulation response can be
compared to the estimated sinusoidal response of Eqn. (2.4-5), and it can also be compared
51
to the response determined by harmonic balance. The specific roles that the simulation fills
are listed below:
1)
To check for non-periodic behavior.
2)
To check the system time constant and transient response.
3)
To determine the output oscillation magnitude and period for different parameters.
4)
To compare the simulated response to the estimated response.
The simulation integrates the state space form of the equations of motion given by
Eqn. (3.3-1), initially at rest where,
dO
0 = -=-=
dt 0, at t =0.
The simulation is written in PASCAL programming language and uses a fourth order RungeKutta integration algorithm. A Runge-Kutta algorithm is chosen because of its programming
ease compared to finite difference methods. Also, step-size changes can be easily
implemented into this method. The Runge-Kutta algorithm is based on matching a Taylor
series expansion about an initial point, for each of the state variables, to a polynomial written
in terms of the initial values, the stepsize, and the functions representing the derivatives of the
state variables. The coefficients of the polynomial are chosen such that a minimum number
of functional evaluations are necessary per step. The fourth order Runge-Kutta algorithm, as
given by Battin [14], is
1
x = x0 +ýh(ko
0 +2k + 2k 2 +k 3 ) + O(h 5)
kO = f(to,xo)
k
1
1
= f( to + h,xo+12 hko)
1
1
k2 = f(t +Th,xo+h kl )
k3 = f(to+h,xo+hko)
dx
wherdt
where,
-=dt
f(t,x) , x(to) = xo , and t-to = h.
52
It is difficult to choose a fixed value for the step-size, h, to integrate the tuning fork
equations of motion. Since the system oscillates at high frequency, a step-size chosen too
large could lead to a divergent algorithm, and a step-size chosen too small leads to an
excessive number of iterations. For this reason, an adaptive step-size procedure is added to
the simulation program. Since the algorithm makes functional evaluations at the step
midpoint, it is convenient to measure the truncation error by comparing the numerical
estimates obtained form one step size of 2h (call it xl), and that obtained from using two
steps of size h each (call this estimate x2). This error, A, is on the order of h5 since the
algorithm is fourth order. The desired accuracy is defined as A0, where
A = Ix2- x1l and
AO =h (Ymax)
where,
ymax = amplitude of oscillation
Eh = 0.001.
The desired accuracy is defined such that the truncation error is always less than 0. 1%of the
oscillation amplitude. The procedure appropriately scales the step-size at each iteration such
that it is at a maximum value that remains within the error limits. A copy of the simulation
source code is given in Appendix A.
The program is run by interactively entering the system parameters, a specified
transient time, and the number of periods for the stored data. The program integrates up to
the specified transient time without storing data, and then stores data for the specified number
of periods. The discrete data for the output angle and angular velocity, as a function of time,
is printed in tabular form. In addition, the oscillation amplitude and period, the number of
iteration steps, and the ratio of the simulated amplitude to the estimated amplitude of Eqn.
(2.4-5), are also recorded. The program is run on an IBM PC using the Turbo Pascal
compiler developed by Borland International, Inc.
The simulation data gives the approximate system response for different sets of
parameter values. The discrete data for the output angle is presented in graphical form which
53
is compared to the estimated sinusoidal output. Simulations are performed for various values
of e and Q, and at a frequency of 1 kHz. For all cases, the input to the system is a unit step
with magnitude Q = 1.0 rad/sec.
First, simulations are run for various values of the quality factor, Q,while also
varying the transient time input. This determines the time period necessary for the output to
reach a steady state oscillation. The value of the system time constant, Tc, is then
determined. For a second order system, the time constant is defined [33] as the time at which
the output reaches 63% of its steady state value, assuming the system is initiated at time, t =
0. Figure 3.4 plots the ratio of the simulated time constant to the estimated time constant
from Eqn. (2.5-6) as a function of the stability parameter, eQ. The estimated time constant is
given as
2Q
Tc,est = -
In order to check the simulated system response against the estimated linear response
from Eqn. (2.4-5) and against the harmonic balance prediction from Figure 3.3, three
simulation runs are presented here. The system quality factor is fixed at Q = 200, which
represents an acceptable design value, and the frequency is fixed at 1kHz. The data is
recorded after an allowed transient period of 6 seconds, which well exceeds the system
transient period. The three cases are
1) eQ = 1.0
within estimated stability region.
2) eQ = 4.0
near estimated stability boundary.
3) EQ = 16.0
exceeding estimated stability limit.
The results are shown in Figures 3.5 - 3.10. The phase plane plots give a
visualization of the periodic nature of the output. In the figures, the simulated output is
compared with the estimated sinusoidal output. It is also compared to the first harmonic
oscillation curve obtained by using the Fourier coefficients of the preceding section. The first
54
two plots show an almost exact correspondence between the three predicted response curves.
However, the three curves deviate in the third plot. For the parameter values of this case,
which exceed the estimated stability region, the periodic terms in the equation of motion result
in a substantial phase shift. It appears from the plot that the phase shift is nearly that
predicted by the harmonic balance,
)pred = 0.71 radians.
However, the simulated oscillation magnitude exceeds the harmonic balance prediction
magnitude by 24 %. The discrepancy is probably due to errors in evaluating the Fourier
coefficients for a system with a large stability parameter by a 4X4 matrix. As discussed in
Section 3.3, the 4X4 truncation is only valid for small values of the stability parameter where
the Fourier series converges rapidly.
The simulation leads to the following conclusions:
1)
The periodic terms do not significantly affect the system transient response, as seen
from Figure 3.4.
2)
When the stability parameter is within the estimated stability limit, the system output is
accurately approximated by the sinusiodal response of Eqn. (2.4-5). This is seen from
Figures 3.5 and 3.7.
3)
For values of the stability parameter exceeding the estimated stability limit, the output
magnitude decreases and a phase shift occurs, as seen form Figure 3.9.
4)
The phase shift that ocurs for high values of eQ is approximately that predicted by the
harmonic balance. However, the magnitude of oscillation for high values of eQ is
greater than that predicted by harmonic balance, as seen in Figure 3.9.
5)
2n
The simulated responses are all periodic with period, T = -co
55
Ratio of Sim. Time Const. to Estimated
Time Const vs. Stab. Parameter
1.2
0.8
0.6
0.4
0.2
0.
0.
5
10
15
stability pararneter
Figure 3.4
20
25
eQ
Simulation Results For System Time Constant
30
EQ
0.00020
-
10
0.00015
0.00010
0.00005
0.0000
-. 00005
-. 00010
-. 00015
-
nnn0n
.6000 .6005 .6010 .6015 .6020 .6025 .6030 .6035 .6040 .6045
TIME (SEC)
simulated response,
Omax = 0.000160 radians.
estimated response from neglecting periodic
terms,
ee = 0.0001592 sin ((ot).
predicted response from harmonic balance,
Ohb
Figure 3.5
= 0.0001592 sin (cot + 0.00332).
Simulation results for e = 0.005
Q = 200
o = lkHz
EQ
1.0
=
1.5
1.0
LL)
(I)
0.5
0.0
-0.5
-1.0
- C;
-. 0002
-.0001
OUT.
Figure 3.6
0.0001
0.000
ANGLE
Phase Plane Plot for
0.0002
(RAD)
E = 0.005
Q = 200
o = 1 kHz
EO
4.0
=
0.0008
0.0006
0.0004
0.0002
0.000
-. 0002
-. 0004
-. 0006
-. 0008
.6000 .6005 .6010 .6015 .6020 .6025 .6030 .6035 .6040 .6045
TIME
(SEC)
simulated response,
emax = 0.000644 radians.
estimated response from neglecting periodic
terms,
ee = 0.000637 sin (wt).
predicted response from harmonic balance,
ehb = 0.000638 sin (cot + 0.053).
Figure 3.7
Simulation results for e = 0.02
Q = 200 (o= l kHz
=
eQ
4.0
5.
4.
3.
2.
1.
0.
-1.
-2.
CD
<
-3.
-4.
-5%
-.0008 -.0006 -.0004 -.0002
OUT.
Figure 3.8
0.000
ArNGLE
Phase Plane Plot for
0.0002 0.0004 0.0006 0.0008
(RAD)
e = 0.02
Q = 200
o = 1 kHz
r
EQ
rrrrr
1650
-
U. UUJ
0.002
0.001
0.00
-. 001
-. 002
-
nn0
.6000
.6005 .6010 .6015 .6020 .6025 .6030 .6035 .6040 .6045
TIME
(SEC)
simulated response,
emax = 0.002413 radians.
estimated response from neglecting periodic
terms,
ee = 0.002548 sin (ot).
predicted response from harmonic balance,
Ohb = 0.00194 sin (cot + 0.71).
Figure 3.9
Simulation results for e = 0.08
Q = 200
(o = l kHz
EQ
=
16.0
15.
10.
C
0.0
FC)
-5.
I
..J
-10.
-15.
-20.
-. 003
-.002
-.001
OUT.
Figure 3.10
0.00
0.001
0.003
(RAD)
ANGLE
Phase Plane Plot for
0.002
E
= 0.08
Q = 200
o = 1 kHz
3.5 Stability in Terms of Floquet Theory
The simulation results are encouraging. They verified that within the estimated
stability region, the gyroscopically forced output is accurately approximated by neglecting the
periodic terms in the equation of motion. However, the simulation results can not be used to
verify that the system is indeed stable. It is possible that an unstable system can have a
periodic output, that does not appear to grow with time, if it is forced periodically
[11].
Since the gyroscopic torque is periodic, an alternate means for verifying the stability of the
system is necessary.
The most common method of determining the stability of an LP system is the use of
Floquet Theory. The celebrated Floquet Theorem states that if the solution to a periodic
differential equation is known for one period, then it is known for all time. The Floquet
Theorem is also expressed in Eqn. (3.2-9) which shows that the solution to a periodic
equation is related to the solution one full period away by a complex constant. The state
space form of the tuning fork equation of motion (3.3-1) can be written in terms of a state
transitionmatrix ,
a (t) =
d
d
(t,to) x (tO)
(t) = G
(t,to)
where,
(3.5-1)
and ! (toto) = I .
(3.5-2)
Because of the Floquet Theorem (3.2-9), a discrete transitionmatrix can be defined which
describes the solution at discrete points sampled every period, T, from an initial point x(0),
x (T) =
(T)a (0)
and
x (kT) = !k (T) x (0) where I (T) = ( (T,0) = q (t+T,t).
63
(3.5-3)
Eqn. (3.5-2) indicates that the solution to LP systems can be written in discrete-time
form. The stability criteria for a system of this form depends on the eigenvalues of the
discrete transition matrix, and is written as a theorem [11].
Theorem 3.1:
The naturalresponse of a periodically time-varying system is stable if and only if n
eigenvaluesof the discrete transitonmatrix have magnitudes greaterthan one and that an
eigenvalue with unity magnitude is not degenerate.
However, it is not trivial to determine the discrete transition matrix that satisfies Eqn.
(3.5-1) for the plant matrix, G, defined in Eqn. (3.3-1). The most convenient method for
determining I (T) is to numerically integrate Eqn. (3.3-2) for a given plant matrix. Then the
calculated eigenvalues of the state transition matrix can be used to check the stability of the
system for a given set of parameters. But, instead of implementing another numerical
integration scheme for Eqn. (3.3-2), the simulation of the previous section can be used to
obtain the same information.
The homogeneous equation of motion is integrated with the initial conditions
perturbed from equilibrium. Then, sampling the output at integer multiples of the period
gives information about the placement of the discrete transition matrix eigenvalues. If the
output for both states decreases in magnitude each period, then the eigenvalues have
magnitude less than unity. On the other hand, if the output increases in magnitude for either
state, one of the eigenvalues has magnitude greater than one.
The initial conditions are perturbed from a state of rest such that,
9 (0) = 0.05 radians, and
d
0 (0) = 0.1 rad/sec.
The system parameters e and Q are varied, and the discretely sampled output for the
perturbed system is recorded. For all values of the stability parameter within the estimated
stability range, the eigenvalues are concluded to have magnitude less than unity. A plot of the
64
discrete output for eQ = 4.0 is shown in Figure 3.11. Moreover, the system was checked
for robustness, and values up to eQ = 30.0 displayed stable responses.
response to initial value perturbation
at eQ
4.0
-
0.06
0.05
a
x
0.04
0.03
0.02
0.01
t•
00
300
100
400
period number
Figure 3.11
Response to Initial Value Perturbation for
E = 0.02, Q = 200, and to = 1 kHz
65
500
4
SYSTEM MATHEMATICAL MODEL
The process used to design the micromechanical tuning fork gyroscope is
illustrated in Figure 4.1. The goal is to develop a reasonable design and a
corresponding set of performance predictions. The design is classified as a baseline
design. This means that it represents a starting point for the development of a
micromechanical tuning fork instrument, and not necessarily the exact design for a first
prototype. The performance predictions are used to evaluate the potential of the device
as an angular rate sensor. It is important to note that the performance predictions are
only analytical approximations. They are intended only to be accurate enough to make
conclusive comparisons with similar angular rate sensors.
The first two chapters of this paper provided the concept demonstration and
preliminary configuration steps in the design process. Chapters 2 and 3 served to
analyze the system dynamics. An appropriate linear model for the gyroscope sensitivity
was derived, and verified, for a given range of the system parameters. The next step is
to develop a thorough mathematical model of the system. The model relates the device
dimensions and input parameters such as driving voltage, frequency, temperature, and
pressure, to the gyroscope performance. The performance of the device includes the
sensitivity to an applied angular rate about the sensitive axis, the mechanical limitations,
robustness, electrical output, damping levels, and identified error levels. By
thoroughly modeling the system, a feasible design can be produced by varying the
dimensions and input parameters until a suitable performance level is obtained.
66
CC
in
preliminary
configuration
dimensions
system input
parameters
sensitivity
error levels
PREDICTED
PERFORMANCE
Figure 4.1
Design Process
DESIGN
4.1
Motor Model
The gyroscope motor consists of the two tines, the two attached proof masses,
and the electronics which apply a voltage across the tines. The mathematical model of
the motor relates the mechanical gain, e, to the applied driving voltage, as shown in
Figure 4.2. The first step in developing the model is to determine the electrostatic force
which vibrates the tines as a function of the applied voltage.
It is clear that the tines cannot be accurately modeled as two parallel plate
capacitors. The width of the tines is not negligable compared to the height. The
applied voltage causes a uniform charge density to accumulate along the top and bottom
(x-y plane) of the tines. Electric flux is radiated from this area and a fringing electric
field is produced, as shown in Figure 4.2. The fringing flux adds to the parallel flux
radiated form the inside of the tines to produce a electrostatic force between the tines.
For the parallel plates alone, the force between the tines is determined by differentiating
the electric energy with respect to the gap distance. The expression for the force is
given [15] as,
Fl =
2
E0 A1 Vd
2 d2 2
2d
Al = hL
c0 a
where,
(4.1-1)
area of inside of tines
permittivity of the gap air .
The electrostatic force due to the fringing electric flux is not determined as easily
because the flux is not one-dimensional. The problem is to determine the capacitance
between two flat plates lying in the same plane as a function of the plate dimensions.
The electric energy can then be determined which is differentiated to give the
electrostatic force.
68
Vd
mechanical gain
driving voltag
gap between tines
-*1 d I.z
widi
tine
it of tines
length
of
tines..
tine vibration
lux lines
1parallel plates
ux lines
>m fringing
L~s
Vd sin cot
Figure 4.2
Diagram of Tuning Fork Motor
The electric flux lines are assumed to be two-dimensional as drawn in Figure
4.3. It is also assumed that the gap distance, d, is constant. This approximation is valid
only if the tine vibration amplitude is small compared to the gap distance. It is
desireable to make a transformation from the z-plane to the w-plane which would
produce a one-dimensional flux bounded by known potentials. An appropriate
transformation would be to transform the upper half of the z-plane into a closed
rectangle with the sides at the potentials V 1, -V0, 0, and V0, as drawn in Figure 4.3.
The value of the potential field in the same plane as the top of the tines, and outside of
the tine gap, is assumed to be constant and denoted by VI. The symbol V0 is used to
denote one-half of the driving differential voltage, so that,
V0 =
1
Vd .
A transformation that maps a line segment on the real axis in the z-plane to a
closed polygon in the w-plane is known as the Schwarz-Christoffel transformation
[16]. The transformation is of the form,
C (z- u) 1(z- U2) ... (z-un)
dw
dz = 1
(4.1-2)
This folds up the real line in the z-plane at the points zr = ur by an angle given by ar,
where,
r = 7(1 +Pr).
The constant, C1 , is known as the expansion constant and is determined by the
particular boundary conditions of the transformation.
To form a rectangle with the four corners given by the points A, B, C, and 1)
from Figure 4.3, the transformation parameters are,
al = a2 = a3 = a4 = y,
u=
--(
d+
) , u2 = -
d,
u3 =
Combining equations (4.1-3) and (4.1-2) gives,
70
d,
and
u4 =
d + w.
(4.1-3)
Z Plane
VO = 1/2 of driving voltage
V1 = assumed constant voltage
outside of tine gap
C
A
V1
I
-a Vo -b
Vi
D
v
I !
|
Vo
-a - V0
W Plane
A
V1
Vi
-Vo
D
~Pf
Vo
I
Iw
V =
0
Figure 4.3 Conformal Mapping for Electrostatic Fringing Flux
w(z) = C 1 (z +2•
Let a =
d
+ w, and b =
w(z)
(z- 6 -
w) (z+d)2
d
w) dz
2(4.1-4)
then Eqn. (4.1-4) is written,
f
=
z-a 2 z2 - b2 .
(4.1-5)
z
b
By making the change of variables C = - , and defining k = - the integral in Eqn.
S
(4.1-5) can be written in Jacobi's form of the elliptic integralof thefirst kind [14],
w (() = C 2
fJ~d
(4.1-6)
The constant , k , is known as the elliptic integralmodulus , and the constant C2 is
defined as,
1
C2 = -. Cc
1.
The constant of integration from the indefinite integral of Eqn. (4.1-6) can be
made zero if the transformation is defined such that the origin in the z-plane maps to the
origin in the w-plane. As the variable z progresses from the origin to pt. C from figure
4.3, the variable Cgoes from r = 0 to
w ()IpLC
= 1.
1 The transformation at C is then,
dC
= C2
o
=2 C2 K (k) .
e- d I -k
(4.1-7)
K (k) is defined as the complete elliptic integral of the first kind. An elliptic integral is
complete. when the integration is taken from zero to unity and the integral is written in
Jacobi's form. Efficient computational algorithms are available [14] to evaluate
complete elliptic integrals. In order to describe the transformation integral at the
complex points 1 and A, a conjugate to the complete elliptic integral is defined [16],
72
dt
K' (k)
(4.1-8)
where t is a complex number. It is shown by Gibbs [15] that the integral conjugate
can be written,
K'(k) = K (k)
where k' =
1 - k2
and k' is called the modulus conjugate. For the line segment CID in the z-plane the
1
variable z goes from 1 to F, and the integration variable in the w-plane proceeds in
the direction of the imaginary axis. The transformation at D is then recognized as,
w IPt. 1
=
C2 K (k) + j C2 K' (k)
=
C2 [K(k) + jK(k')].
By using negative values of z as it progresses from the origin in Figure 4.3 to points
A and B, it is readily shown that,
w Pt. A
w
= -C2K(k)
IPt.
=
and
-C 2 [ K (k) + j K(k') ].
(4.1-9)
The solution to the complete elliptic integral of Eqn. ( 4.1-7 ) can be written in
terms of JacobiElliptic Functions [15,16],
w (z)
=
1
zb
1 sn-l (b
)
(4.1-10)
Using the boundary conditions for the voltages from Figure (4.3) at point D,
K (k) + j K (k') = C2 ( V0 + VI )
C2
aK
and
K (k)
V
which implies
=
(k') VO
K (k)
(4.1-11)
Eqn. (4.1-10) is now written,
w (z)
=
sn-K(k)
(
)
(4.1-12)
The transformation can be checked by choosing points and using the appropriate tables
to evaluate the elliptic integrals and elliptic functions.
73
The electrostatic problem in the w-plane is one-dimensional and can be solved
routinely. Let Cw denote the capacitance between the parallel plates represented by the
segments AB and CI, then
Cw
wL
dW
=
where,
(4.1-13)
CD,
=
the length of the segment CD in the w-plane
dw
=
the distance between the parallel plates in the w-plane
L
=
The length of the tines
From the mappings given in Eqn. (4.1-9) this can be written in terms of z-plane
dimensions,
C = Cw
E0 L K (k')
2 K (k)
=
(4.1-14)
Let U denote the electric energy between the parallel plates and F2 denote the
electrostatic force between them,
U2 =
F2
1-4CVd2
C Vd
2
2
Vd2
I a
ad
(4.1-15)
where d is the gap between the tines as shown in Figure 4.2. Note that the electric
energy is written in terms of the differential voltage between the plates, denoted by Vd.
Differentiating the capacitance from Eqn. (4.1-15),
[a
aC
L Pd
ad
2
K (k) -- K (k')
-K
(k')
a
K (k)
ad
K (k)
Let the integrands for the complete elliptic integral (4.1-7) and its conjugate be denoted
by the functions g (ý,d) and f (ý,d),
K (k)
=
g (C,d) dC,
K (k')
=
f (ý,d) d( .
o
74
(4.1-17)
The values of the modulus and its conjugate are written in terms of the gap distance and
tine width,
k =-
b
d
= d+ 2w
k'
1
and
d2
w)
2
(d + 2w)
(4.1-18)
By Leibnitz's Rule [18] the partial derivitive with respect to the gap distance can be
taken inside the integral so that
a
-f(ý,d)
ad
dC
fg (0,d)
DC
L
ad
2
d
-
f ( ,d) d
a
0
0
a
-g (,d) dý
Iad
2
Ig(d)d
g(C,d) dC]
(4.1-19)
Substituting from (4.1-7) and (4.1-18) and differentiating,
a f( ,d)
ad = Al (d,w)
a g(ý,d)
= A2 (d,w)
ad
Al (d,w) =
A2 (d,w) =
2w
2
(d + 2w )
2wd
(d + 2w )3
Substituting into Eqn. (4.1-19),
75
2.(2
2)
1
-
22 )
(2)2
2 1 - k'2 2)2
where,
and
(4.1-20)
ac
ad
co L K (k) Al (d,w)
-2
2K2 (k)
2
K2
1C(I
(1
K-
Eo L K (k') A2 (d,w)
d_
(2)~
(
-
k'2) 2 )2
2 dý
(k)
2),
1-(22
1- k22)2
-
0
((4.1-21)
The integrals in Eqn. (4.1-21) can be shown to be the derivative of the complete
elliptic integral of the first kind,
dk-
dk
f
2)- ( d
1I
(4.1-22)
One of the most significant results in the study of elliptic integrals came from the
famous French mathematician Adrien-Marie Legendre in a paper of 1826, and it may be
stated as a theorem [14],
Theorem 4.1:
If P(x) is a polynomial of at mostfourth degree with real coefficients and if R is
a rationalfunctionof two variables with real coefficients, while x is restrictedto a range
in which P(x) is positive, the integral
f R[x,
P(x)]
dx
can be expressed as a linear combination of terms, each of which is eitheran elementary
function, or an elliptic integralof thefirst, second, or thirdkind.
76
dK
Theorem 4.1 implies that the derivative, K , can be written in terms of elliptic
integrals. The relationship was derived by Bowman [17] and is given by,
E (k) - k' 2 K (k)
d K (k)
dk
k'
-
2
(4.1-23)
The term E (k) represents the complete elliptic integralof the second kind, and it is
also readily evaluated by computational algorithms.
Substituting Eqns. (4.1-23) and (4.1-21) into (4.1-15) for the electrostatic force
from the fringing,
2
Eo
V2L
4kk'
E (k') A,
k K (k)
k K (k') At
K (k)
E (k) K (k') A
k'K
K2(k)
k' K(k') A
K (k)
J(4.1-24)
The total electrostatic force that the tines are subjected to is obtained by adding the
forces from the parallel flux (4.1-1) and the fringing flux,
FO = F 1 + 2F
2
.
(4.1-25)
To include fringing from both the top and bottom of the tines a factor of two is
included. The computational techniques for evaluating Eqn. (4.1-24) will not be
described in this paper. The reader is referred to Battin [14] and Press et al [19] for a
description of the algorithms used. For a set of typical dimensions for the tines,
L
=
500 microns
d
=
5 microns
h
=
15 microns
w
=
4 microns
The force component due to the fringing is calculated to be 20 % of the total force.
The fringing has the advantage of adding a significant amount of force per volt, but has
the disadvantage of being highly nonlinear.
The electrostatic force represented by the equations above produces a uniform
load on each tine. The tines are modeled as Bemoulli-Euler beams fixed at both ends,
as shown in Figure 4.4. The static deflection curve is given by [20],
77
x - axis
y - axis
/
--
_L1._._
k sensitive axis
IL
Fo / L
Om
uniform electrostatic load
Ot
/1
~
j
mt
El
u (y,t)
deflection curve
JI
=0
Figure 4.4 Tine deflection from electrostatic force
)
u ()
=
Foy
2
24EI(L(4.1-26)
where I denotes the beam inertia of the tine,
I = TWh w3
Since the tines are assumed to be driven at a frequency significantly below their
resonance, as described in Chapter 2, the deflection curve is sinusoidal at the driving
frequency, wd
u (y,t) = u (y) sin ct.
If the proof mass is approximated by a point mass with the magnitude denoted
by, mp , and the inertia of the proof mass about the y-axis (sensitive axis) is denoted by
Im ,
Im = m rom + u
,t
where,
rom = nominal distance from the proof mass to sensitive axis
Assuming that the tine vibration amplitude is small such that second order terms in
u(y,t) can be neglected, the proof mass inertia becomes,
Im = mprom + 2 mp romu 2, t = Im + Ilm sin ot
where, from Eqn. (4.1-26), the oscillating component is given by,
3
F0OL mp rom
m 192
El
(4.1-27)
For each tine of mass mt the moment of inertia about the y-axis is defined as It,
where,
It =
[rot + u (y,t)] 2 dy
(4.1-28)
Again assuming that second order terms in u (y) are negligable and that the height and
width of the tines are small compared to the length, Eqn. (4.1-28) becomes
It = mrot + 2 phwrot
79
u(y) dy sinat
where,
rot = nominal distance from tine center of mass to the y-axis
p = density of tine material (silicon)
Integrating the deflection curve of (4.1-26) and writing the tine inertia about the y-axis
as a nominal term plus an oscillating term,
where
It = I0t + Ilt sin wt
IOt = mt rot2
and
mt r0 tF 0 L3
I1
360 El
(4.1-29)
.
Combining Eqns. (4.1-27) and (4.1-29) the mechanical gain of the device is a linear
function of the electrostatic force,
F0 L
3 mp rom
El1
192
mt ro+
360
2
2
mprom + mtrot
80
(4.1-30)
4.2 Sensor Model
The function of the sensor is to transform the magnitude of the output angle vibration
into an electrical signal. The sensor is comprised of four capacitance plates on each side of
both wings. Four electrodes are situated above the plates, and an excitation voltage is applied
across the four sets of capacitor plates. As the wings vibrate due to the gyroscopic torque the
capacitance between the plates is altered. The electrical signals coming from the capacitors
are combined and processed to provide the gyroscope output signal.
This is the same type of sensor used on the double gimbal micromechanical
gyroscope designed by Boxenhorn [1,2]. The sensor on the Boxenhorn gyroscope has been
shown to be sufficiently linear. For simplicity purposes, the signal processing electronic
configuration is designed to be identical to the open loop configuration of the Boxenhorn
gyroscope. In this section, only the fundamentals of the sensor model are presented. The
reader is referred to Boxenhorn for a more detailed derivation.
Figure 4.5 shows the sensor geometry on one wing. The parameters xl and x2
denote the distances from the sensitive axis to the front and back edges of the sensor plates,
respectively. The distance, g , is the nominal gap between the capacitor plates, and Wy
denotes the width of the plates (y-dir) . On one side of the wing the capacitance is increased
by the angular rotation, 0, and is given by [1,15]
e WY dx
Ci
gg-0x
- ( xl
CoWy Ig
9g-0x2
(4.2-1)
And, on the other side the capacitance is decreased,
Cd
F
WY dx
g+Ox
Co Wy In
+ exl
9 +g+
x2
Expanding Eqns. (4.2-1) and (4.2-2) in a Taylor series about the point 8 = 0,
(4.2-2)
C
2
o-w 2g(x
2
=
+
-
g2
2
6g2W(x3+ 0(0 3)
x)
6 g2
where Cthe
represents
nominal capacitance between the parallel plates,2
2g2
6g2
where CO represents the nominal capacitance between the parallel plates,
+ O (0 3)
o0 Wy (x2- xl)
g
(4.2-3)
If the capacitances are subtracted,
AC = Ci- Cd = KsCO 0 + 0(03)
where the sensor scale factor is given by
Ks
xl
+ x2
22g
x2
(4.2-4)
Subtracting the two variable capacitance signals has two advantages. First, the error
in the linear model of the sensor is reduced to O (03) since the quadratic terms in the
expansion cancel each other. Second, any error caused by a deflection of the wings in the zdirection will be eliminated. The deflection will cause a change in capacitance of equal
magnitude on both sides of the sensor, and the errors will cancel each other.
A diagram of the signal processing electronics is shown in Figure 4.4. The AC signal
from each wing is added, amplified, demodulated twice, and run through a low pass filter to
produce the voltage output. The baseline design assumes that the electronic configuration is
fixed. The relationship between the output signal and the change in capacitance is given by,
eout = 100 (2)2
Cfb
Ex
.
7(4.2-5)
suspended electrode plates
,I
x
/Y
ýN
Y
~1§1J~
47~
w Illy
Z -axis
y -axis
x -axis
C
i
II
AC
AC
Cdd
d
Q--II
1
R =
1.0 E8 ohms
fb
Figure 4.5 SeFnsor Diagram
4.3 Mechanical Factors
The gyroscope rotation is provided by two thin, rectangular silicon shafts which
twist axially about their cross-sectional center of gravity and serve as torsional springs.
The flexures extend on both ends from the wings and are fixed to the substrate. They are
designed assuming a fixed length and width, and varying the thickness to yield a torsional
stiffness that corresponds to the desired frequency.
It is important that the stiffness model for the flexures is accurate since the driving
frequency is tuned to the torsional resonant frequency. The classical St. Venant solution
which determines the stiffness of the flexures [21] is not applicable in this case for two
reasons. First, the St. Venant solution assumes free ends, that is, it allows the rectangular
ends to warp. But, in this case, both ends are fixed and constrained from warping.
Second, since the flexures are of a different thickness than the wings, they must be
fabricated using separate etching diffusions. When boron is diffused into the silicon,
which is needed for the selective etching process, the boron atoms replace the larger silicon
atoms and cause a lattice contraction. It is the relative contraction between the wing and the
flexure that produces an axial pre-stress on the flexure which affects its torsional stiffness.
The fabrication process will not be discussed in this paper. The reader is referred to the
literature [22,23] for descriptions of micromechanical fabrication processes.
The flexure stiffness problem was solved by J. A. Connally in his MIT Master's
thesis [24] Torsion of a Thin Rectangular Beam with Axial Prestress and Ends Constrained
from Warping. The stiffness equation developed by Connally is used in the design
presented here. The torsional stiffness for the flexure shown in Figure 4.5 is given by,
kt -
3 fzfx 3 G
lIf- Le
where,
L C =zE
12 (o+ G+]G
(4.1-1)
where,
00 = the axial prestress on the flexure
1
=
length correction factor
fz =
fx
thickness of flexures
=
width of flexures
Connally's equations were verified by a finite element model, and the error was
shown to be within 4 %. Also, the model was used to calculate the natural frequencies for
a double gimbal micromechanical test device designed by Boxenhorn. For six different
frequency tests, the Connally model yielded accuracies within 10%.
The magnitude of the prestress is determined from the relative diffusion contraction
between the flexures and the wings. The misfit factor between boron and silicon atoms is
defined as [25],
=
i
where
'si
(4.3-2)
rb = atomic radius of boron = 0.98 angstoms
rsi = atomic radius of silicon
==si Poisson's
= 1.17 angstroms
ratio for silicon = 0.33
The axial contraction of each flexure (y-dim) is proportional to the misfit factor, the
concentration ratio, and. the axial length of the flexure [25]
8f = (Cbf) If
csif
and for the contraction of each wing,
SW =
[(
w) wy
(4.3-3)
where,
cbf = concentration of boron atoms for each flexure
csif = concentration of silicon atoms for each flexure
cbw = concentration of boron atoms for each wing
csif = concentration of silicon atoms for each wing
Generally, for fabricating micromechanical structures with thicknesses less than 10
microns deep, the boron concentration is two orders of magnitude less than the silicon
concentration. The resulting axial stress in each flexure is given by,
o
=
S(Sw - 8f))E
If
(4.3-4)
One complicating factor in calculating the torsional stiffness of the flexures is that
single crystal silicon is anisotropic. This means that the elasticity of a single crystal silicon
structure depends on the orientation of the lattice surfaces. The values for the silicon
elasticity constants used in the calculations are
E =
1.7 Ell N/m 2
and
G =
5.1E10N/m 2
These values were documented by Connally [24] by assuming that the etching process will
cut symmetric structures along the (1,1,0) crystal planes.
Equations for the shear stresses induced in the flexures by the axial twisting were
also determined by Connally. They are shown [24] to be at very low levels for
micromechanical structures, and hence are not of much concern. However, if the width of
each wing (y-dim) is designed to be significantly longer than the length of each flexure, the
axial stress (4.3-4) becomes very large. This tensile stress stiffens up the flexures and
reduces the reliability of the device if it approaches the yield strength.
bz is the
thickness of
the
:rate
Wy
wing
b
y
strain
beam
substrate
z/
•
,
I7 -
x -axis
,i
aYis
figure 4.6 Torsional Flexure
y - axis
To reduce the axial stress, the device is designed with a strain relief beam in the
substrate near the flexure end. The strain relief beam is a slot which is cut through the
substrate and absorbs some of the strain induced by the lattice contraction. It is shown in
Figure 4.6. The flexure and the strain beam act as linear springs in series acted upon by the
tensile deflection. The effective linear stiffnes is given by
keff
=
kf =
kl
Sklf
klf klb
Ebc
klb= 192bxEIb
Ib =
where
where
(flexure linear stiffness)
(strain beam linear stiffness)
bz by 3 .
(4.3-5)
By appropriately designing the dimensions of the strain beam so that the beam stiffness is
much less than the flexure stiffness, the majority of the contraction strain can be absorbed.
Stress levels in other areas of the device are not of much concern. For reasonable
levels of external shock and vibration the possibility of structure failure is very low.
Silicon is a very high strength material, and the device is homogeneous such that there are
no obvious stress concentrations areas. The equations for the tine twisting, flexure
bending, and wing bending, developed in Section 2.2, are also implemented in the system
model to verify the rigid body assumption for a specific design.
4.4 Damping Model
In the system model introduced in Chapter 2, it was assumed that the torsional
vibrations encounter a restraining damping torque that is proportional to the angular
velocity. The source of the damping torque is almost wholly attributed to fluid damping.
Since the device is homogeneous with no joints or sliding surfaces, any structural damping
is assumed to be negligable. The damping is an important design consideration since the
sensitivity (2.4-5) and the stability criteria (3.2-19) are both written in terms of the system
quality factor. In fact, the Boxenhorn micromechanical gyroscope [1] is designed to be
operated in a vacuum package to limit the air damping and increase the sensitivity. For the
tuning fork configuration the air damping is confined mainly to the sensor wings so that the
damping is less significant than the double gimbal configuration.
The damping is written in terms of the quality factor, Q, which is related to the
system frequency, inertia, and linear damping coefficient by Eqn. (2.4-5). The
mathematical model of the damping is intended to provide an approximate upper bound on
the damping torque, or a lower bound on Q. This is meant to simplify the fluid mechanics
analysis yet still provide a conservative estimate of the gyroscope sensitivity.
The oscillating wing induces flow along its surface which produces a pressure
difference between the two sides of the wing. The pressure difference leads to a damping
torque that resists the oscillation. Two flow patterns that provide the majority of the
damping torque are identified:
1) air flow impingement on the wing surface as it oscillates.
2) air pumping through the channel between the wing and its bridge electrode.
The damping due to each of these flow patterns is analyzed seperately, then it is multiplied
by two to account for both wings to provide the system Q estimate.
Air Flow Impingement
The air flow on the oscillating wing is modeled as a plate suspended in a free stream
of air with velocity
U,
Wx
X
t
=
Wx
=
length (x-dim) of the entire wing
=
co
where
angular velocity of wing in rad/sec
(4.4-1)
This free stream velocity corresponds to the maximum linear velocity on the oscillating
wing. The model is simplified by assuming that the flow over the entire wing is in one
direction. Actually, as the wing rotates about its pivot, each side of the wing is moving in a
different direction. However, for the simple model presented here, the damping effect is
equivalent. A diagram of the model is shown in Figure 4.7. The viscous air flow is
assumed to be incompressible and steady with respect to the oscillating wing. Also, the
flow is assumed to be two dimensional such that the air flows along the. plate in the xdirection as drawn. Again, this is conservative since the neglected flow around the plate
edge in the y-direction will decrease the pressure gradient. The two dimensional continuity
equation in differential form is written
au
-+
03x
aw
-
az
0
(4.4-2)
where,
u = air flow velocity component in x-direction
w -
air flow velocity component in z-direction.
The Navier Stokes equations [26] in two dimensions describe the flow momentum
u-
au+
ax
u-
aw
ax
w
au
az
+ w-
aw
az
ap
-
+
Pax
-
1
ýL +a2
U+
P
P
P aZ
+-
t
a
a2
2iw
P a x2
U
a Z2
x2
+I
a2W
2
(4.4-3)
The density of the air, p , is assumed constant , and the viscosity is given by gt. For a noslip condition along the surface of the wing the boundary conditions are
w = 0
at
z = 0,
-L< x < L
u = 0
at
z = 0,
-L< x < L
w
= - U. , and
u
=
0
at free stream.
This problem was solved by Happel [27] and the solution for the resultant force on the
plate is given by,
Ff = -16 tLUc
= -16 t L 2 c
for L =
(4.4-4)
Translating this to a pressure drop and substituting the dimensions of the wing
Wx
A Pf = 4coW
.
(4.4-5)
Channel Pumping
As seen from Eqn. (4.2-4) the gyroscope output signal is inversely proportional to
the gap distance between the wing and the electrode plate suspended over the wing. For a
small gap, as the wing oscillates air is pumped between the electrode plate and the wing.
The viscosity of the air flow adds to the pressure differential between each side of the
wing.
Again, the flow is assumed to be two dimensional, as shown in the bottom half of
Figure 4.7. For the pumping effect only one-half of the wing is analyzed. This is because
the bottom of the substrate beneath the wing is at a great enough distance such that no
pumping occurs. It is furthur assumed that along the oscillating axis (x = 0) the air is
unperturbed.
Define Qf as the volumetric flow rate through the channel
Qf = -L ( gL Wy)
(4.4-6)
The dimension Wy is the width of the wing. For small values of the oscillation
magnitude, 8 , at an instant in time the channel gap is a function of x,
g(x) = go -
x.
(4.4-7)
Equation (4.4-6) is then
Qf (x)
L Wyx
=
.
(4.4-8)
du
Making the approximations U = 0 and w = 0 the Navier Stokes equations
(4.4-3) reduce to
1 dP
dx
d2u
(4.4-9)
Solving gives
u
1 dP
2gLdx
For a no-slip condition at z =
u(x,z)
2
- and z = -
I dP
= 2
(4.4-10)
+ Clz + C2.
dx
z2
, the constants are evaluated to give,
2
)
(4.4-11)
Define Q'f as the profile flow rate where
so
Q'
3
u(x, z)dz
=
dP go
dx 12 g
(4.4-12)
Then the volumetric flow through the channel is
dP
=Ydx
Qf = WyQ'f
3
go
12
12 C
(4.4-12)
Equating Eqn. (4.4-12) and Eqn. (4.4-8) gives an expression for the pressure gradient
along the plate,
S-
12gtLox
dx
3
(4.4-13)
Eqn. (4.4-13) is integrated and boundary condition P = Patm at x = L is used to
evaluate the constant of integration
P (x)
6 gLo
3
90
(L 2 - x2)
+ Patm
(4.4-14)
Flow imoingement of one wing
sensor wing
S-L
'-I.
Wx
x=-L
x=L
atm
Channel Pumoina
electrode plate
Jd •
l
i~ m m l
n
•
.,:...,:.. .......................................
x
90
-p
P
Q
ft
x= L
Figure 4.7 Damping Model
93
Satm
Integrating the change in pressure on both sides of the wing to obtain the resultant force
gives
Fp
j (P (x)
- Pa,)
dA
=
(P(x) - Pan) dx
Wy
(4.4-15)
Substituting the dimensions of the wing L =
W
--
yields the result
WN WY4 ) U
3
Fp =- W 4 4go
g03
(4.4-16)
The moment arm for the resultant force on the wing is
4
Thus, the net restraining torque from both flow patterns is given by,
Tdamp =
2Wx (Ff
+ Fp)
(4.4-17)
where the forces are given in Equations (4.4-4) and (4.4-16) and the factor of two accounts
for two wings. The net restraining torque relates to the linear damping coefficient by
Tdamp = bt d-
This can be written in terms of the system quality factor to give
Q =
2 02 10
W (Ff+Fp)
(4.4-18)
It is seen from the force equations that if the gap distance is significantly smaller
than the wing dimensions, the pumping effect will contribute the majority of the damping.
This indicates that if the conservative model from Eqn. (4.4-18) gives a value of Q which is
too small, design alternatives to reduce the pumping should be investigated. Strategically
oriented slits in the wing will reduce the damping from the channel pumping. However,
the baseline design presented in this paper will be restricted to a solid wing design, and if
the damping is excessive, a vacuum package will be implemented.
4.5
Error Sources
It is now appropriated to discuss the various error sources that are inherent in the
micromechanical tuning fork gyroscope, and to determine design criteria to minimize the
errors. The error sources are classified into four categories: modeling errors, external
torques, internal torques, and random noises. This excludes errors due to cross coupling
and time-variant terms in the equation of motion which were investigated in Chapters 2 and
3.
Modeling Errors
The calibration of the gyroscope linear response to an applied rate depends on fixed
values for system frequency, quality factor, and mechanical gain. However, during a
mission the gyroscope is subjected to changes in temperature and pressure. If the stiffness
properties of the torsional flexures are altered, the driving frequency will differ from the
torsional resonant frequency. This will decrease the sensitivirty. Changes in pressure will
affect the air damping properties so that the quality factor is not a constant. Furthermore,
the mechanical gain depends on the tine stiffness properties which are affected by
temperature and fatigue.
It is necessary that the system is designed to be robust to these deviations. The
solution to modeling errors is best approached on a system level rather than an instrument
level. Two possible methods to minimize these errors are packaging techniques and
feedback control. First, it is desirable to isolate the instrument from the environment by its
packaging. This would keep the gyro at a reasonably constant temperature and pressure.
But, the design tradeoff is when the cost of the packaging, relative to the cost of the gyro,
becomes excessive. It is then more efficient to lock the driving frequency on the torsional
resonance through a feedback configuration, and control the motor by feeding back the tine
vibration amplitude. Feedback system and packaging design analysis is not presented in
this paper, but represents a vital design study for a micromechanical tuning fork gyroscope.
External Torques
One of the advantages of a micromechanical gyroscope is its ruggedness. It is thus
important that the device is insensitive to external vibrations and accelerations.
Accelerations normal to the plane of the gyroscope, combined with a mass unbalance, yield
a torque about the sensitive axis. This is a DC torque and will not affect the output signal.
This requires that the mass unbalance is small enough that the maximum possible
acceleration does not cause the wings to rotate to the point of striking the sensor plates.
Micromechanical surfaces have a tendency to adhere to each other upon touching. Also,
oscillating torques from external vibrations should not result in any errors since the system
frequency is designed to be significantly higher than any reasonable external vibration
frequency.
Internal Torques
It is also possible that the vibrating motor will lead to oscillating torques about the
sensitive axis in the absence of an applied angular rate. The output signal of the gyroscope
can be calibrated to account for these zero rate errors , but the gyroscope design should
minimize these errors.
One source of an internal torque error is motor-sensorcoupling. It is possible that
the electric field which drives the tines can also effect the capacitance between the sensor
wing and its bridge electrode. However, for the tuning fork configuration this effect is
minimal. The motor is driven internally so that the parallel electric flux from the motor
should not cross the sensor gap. The motor fringing flux coupling with the sensor can be
minimized by situating the tines close to each other, and increasing the sensor wing area.
The internal motor represents a possible advantage of the tuning fork configuration
over the double gimbal [1] micromechanical configuration. The advantage can not be
quantified because it is difficult to accurately model the motor-sensor coupling. The
magnitude of the coupling is most efficiently determined experimentally. For the purpose
of the baseline design study, it is sufficient to say that the motor-sensor coupling can be
made minimal.
Another source of internal torque is a misalignment in the z-direction of the two
proof masses. As the tines oscillate, if the proof masses do not have their center of masses
aligned with the sensitive axis, an oscilling torque will occur. Figure 4.8 shows the
masses misaligned by a distance Az. If Xm denotes the amplitude of vibration at the tine
midpoint, the torque about the sensitive axis from the unbalance is given by
Tunb = m
z-- (Xsincot)
dt
2
= mpA z C (Xmsin at)
(4.5-1)
Substituting the unbalance torque (4.5-1) into the simplified equation of motion (neglecting
the periodic terms) represented by Eqn. (2.4-5), gives
o+
Q
+ o02
=
-mpAzo
2 (Xmsin
ot)
(4.5-2)
The response to the unbalance torque is determined by solving Eqn. (4.5-2) to give
0 unb
(t)
=
z xt
10
COp
.
(4.5-3)
Az
il
i:l
iiiiiiiiiiiiii
iiiiii
ii~iiii
00ýýý
iiiiiiii
ii:ii~iiiti
mp
tine
tine
-
mas s center
of..
-
m
P
.................
:::i·~:::::'~:
...
..
..
...
........
:::::::tiiiii
............
..
....
....
....
....
.................
...................
·· .`'`:::''''
..
...................
::::::
·
:::::::::::
·
:
..................
::-::
··:.::~~:::
· · · .....
..................
.................
:·:::::
· ......
....
........
:rr::::::
::··:
··
...........
··
ii
1··::
:: .....
...
`' ''....
' ..........
...
...............
* :::
.........
: :::::: I
..........
::::::::::
..........
·::i::i:
-~'':::::::
..........
..........................
...........................
.....
....
..
..
....
::·::·::·::·:
....
....
..
..
.
.
.
:··:·::·:
..
.
.
.
..
.
I
.
..
..
·..
..
..
..
...
.
.
...
..
..
..
...
........
.
..
....
.
..
....
...·:
...........................
:iiiiii:·::·:::
.....
....
.....
....
..........................
:: ···· ....
·· ·.··:·
...........................
:
:
·.
..........................
::·::·::·::·:
·
.....
....
..
......
..........................
:.;:;:
· ......
~::~~::::
~'
~ ~~~:;:
center of
mass
.
.... :::::::::::...·····::iii.··:·······...................
..................
center of
mass
x m sin (O)t
Figure 4.8 Unbalance Torque from Proof Mass Misalignment
The response to the unbalance is seen to be in quadrature with the gyroscopic
response, and the demodulation will prevent the unbalance from resulting in an output
signal. But, the proof mass tolerance must be within the limit
2 Io g
_
maxI0
Az <
xm Q mp
Wx xm Q mp
(4.5-4)
This insures that the unbalance torque will not cause the sensor wing to strike the electrode
plate.
Random Errors
There are two identified noise inputs into the system that will limit the accuracy of
the output signal. The first is random thermal noise, which will be referred to as Brownian
Noise. As the wings oscillate, the thermal energy in the air causes a random movement of
the air molecules which leads to a random torque on the wings. This torque is generally
negligable for conventional scale systems. But, for a micromechanical resonator, the
system energy is small enough that the thermal energy is significant.
The Brownian Noise is modeled as a white noise entering the system at the torque
summing junction as shown in Figure 4.9. The amplitude of the white noise is assumed to
be
2
AB =
(4.5-5)
TabKBbt .
where,
KB =
Boltzmann's constant
Tab = absolute temperature
bt
=
linear damping coefficient
The noise propagates through the system and results in a variance of the output. The
relationship between the Brownian noise and the output signal is
x(t) =
GI(s) AB
where,
x (t) =
G (s)
-
eout (t)
=
2fQ
(output voltage signal)
K Ke B/b
(s + a) (s + B)
(4.5-6)
b = 100
= filter bandwidth
(B =
1
Hz forcalculations)
The constants in Eqn. (4.5-6) are explained in Figure 4.9. The output signal can be written
as the convolution of the noise input and the transfer function [28] written in the time
domain,
= fg (u) n(t - u) du
x(t)
(4.5-7)
Taking the inverse Laplace transform of the transfer function gives
= KKe B
b
gi (u)
e- w e-B(
dr
Ko Ke B [et e-at]
b (a - B)
(4.5-8)
The system is assumed to be stationary such that the variance of the output is
written
02
=
E [x (ti) x (t 2)]
=
E [x2 (t)] .
(4.5-9)
Substituting Eqn. (4.5-7) into (4.5-9) and noting that gl (u) is deterministic
o2
81 (u) gI (v) E [n 1 (t - u) nl (t - v)] du dv
=
gi (u) gi (v) AB 8 (u - v) du dv
(4.5-10)
where 8 (u-v) is the Diracdeltafunction .
Substituting Eqn. (4.5-8) into (4.5-10) gives
(e-Bu
KB
KO
b (a-B)
au-)A
u - v )du dv
(4.5-11)
100
Solving the double integral yields the result
2
22
ABKO
2 K B
2 b2a(a + B)
-
2AB 022KeKB
22
ABK2 KB e -2Bt +
2 bB)
2
e-(a+B)t
(a+ B)b 2 (a- B)2
2
-
22
ABKeKB
2e -2at
2 b2a(a - B)2
(4.5-12)
The steady state variance due to the Brownian noise input is then
2
ass,B
22
ABKO K e B
2 b2 a(a+ B)
(4.5-13)
The effect of the Brownian noise on the system output has some interesting
characteristics. In general, the bandwidth of the low pass filter, B, will be small compared
to the system bandwidth, a. The Brownian noise variance is then inversely proportional to
the square of the system bandwith. Substituting for the system bandwidth (2.5-6), the
amplitude of the Brownian noise (4.5-5), and the dynamics constant b = 10 o , gives
2
oss,B
(4TK
2
2) BQ
4 TabKBKe K3
Io0)
(4.5-14)
Assuming that the capacitance constant, electronics constant, and temperature are all
fixed, then the Brownian noise variance is a function of B, Q, 10, and o. A very
interesting obserbation from (4.5-14) is that the variance is proportional to Q. At first
glance, this seems contradictory. Since as Q increases the damping decreases and the effect
of the random molecular movement is reduced, it would seem that this would reduce the
variance. But, increasing Q reduces the system bandwidth which has a squared effect on
the variance to offset the first order decrease in Brownian Noise amplitude. To minimize
the Brownian Noise error, the system should be designed to have a high system bandwidth
to filter bandwidth ratio. Also, by increasing the size of the gyro, the Browniari error will
be reduced.
101
In addition to Brownian noise, there is an additional random noise input into the
system as shown in Figure 4.9. The amplifier of the signal processing electronics contains
an inherent random noise. The noise input enters the system at the AC signal summing
junction. The amplifier noise is assumed to be white and stationary with an amplitude
denoted by Aa. The exact value of Aa is determined from the amplifier specifications, but
in the calculations presented here the assumed value is
Aa = 3.3 E-28 A2/hz.
This value corresponds to the specifications of a typical amplifier for the electronics
confguration modeled in Section 4.2.
The variance of the output due to the amplifier noise is determined by following the
same steps used for the Brownian noise. Denoting the amplifier noise by n2 the inputoutput relationship is
x (t) = G2 (s) n2 (t)
where,
KO Ke B
K=
G2 (s)
K1
s+B
1
(x Ex
=
(4.5-15)
Taking the inverse Laplace transform of the transfer function,
g2(u)
= KI KeK
e-Bu.
(4.5-16)
Substituting the expressions for Aa and g2 (u) into double integral of Eqn. (4.5-10) , the
expression for the output variance due to amplifier noise becomes
2
Ga
K 2 K 22
KKegKeBB 2 Aa
=
ee2Bv dv.
(4.5-17)
Integrating, and looking only at the steady state term, the variance due to amplifer is
2
assa
K2 K2KeB Aa
=
2
102
(4.5-18)
It is seen that the most efficient methods to reduce the amplifier noise error are to decrease
the filter bandwidth, or to decrease the excitation constant.
103
n1
Brown.
LaL±
i
i
lIr
J.
hz
i
cos (
d
t
low pass
filter
with
bandwidth, B
capacitance
constant
electronics
constant
excitation
constant
out
---
c--
K
-----
ampi rier noise
2
A = 3.3 E-28 A/hz
,-
(excitation frequency and voltage)
=
xx ExX
(Eqn.
K
4.2-4)
2g
K
100 (2/i)E~
e
(Eqn.
4.2-5)
Cfb
Figure 4.9 System Diagram with Noise Inputs
104
2
5
DESIGN
The previous chapters have laid the mathematical foundation upon which a baseline
design is developed. The main difficulty in developing a design of a micromechanical
tuning fork gyroscope is that there exists no precedent. The only test results available to
support the design are from the Boxenhorn gyroscope [1] and the observations from the
demonstration model. For these reasons, the only design tool is the system mathematical
model.
The system mathematical model is implemented into a computer program written in
PASCAL programming language and run on an IBM PC. The design program uses an
inputted set of device dimensions and system parameters and outputs a set of performance
predictions. The program allows design tradeoff studies to be performed interactively. At
this early stage in the design of the gyro, the design program is only a very rough model of
the device. It includes models for the physical characteristics that were identified in the
previous chapters. As the investigation into the gyro design continues, the program will
quickly grow as additional system characteristics are identified.
In this chapter, the performance predictions for a set of baseline design parameters
are discussed. A printout of the program output for the baseline design is listed in Table
5.1. The design is determined considering geometric constraints andperformance
tradeoffs, and they are the basis of the two sections of this chapter. It must be noted that
the design is not rigorously optimized. The tradeoffs are analyzed to the point of producing
an acceptable performance level. Optimization is left for a future study.
105
5.1 Geometric Constraints
The geometry of micromechanical structures is restricted by the fabrication
techniques. The micromachining techniques were originally developed by the
semiconductor, industry to etch circuit lines on microchips, and they were basically limited
to simple, planar geometries. Although the advent of micromechanical structures has
greatly increased micromachining technology, there are still broad limitations on possible
geometric configurations.
It is difficult to exactly specify the geometric limits of micromachining since new
techniques are being invented daily. In this chapter, a set of assumed restrictions are
specified. They are based on state-of-the-art technology and resources available at the
Charles Stark Draper Laboratory. The assumptions are conservative and are derived from
the input of technicians and engineers within Draper's micromachining laboratory.
Figure 5.1 shows a sketch of the geometry of the baseline design. The figure only
shows one half of the device which is symmetric about the gold bar (x-axis). The first
geometric restrictions involve the tines. The ratio of the height of each tine to its width is
assumed to be limited to
(h: w)
<
15:4 .
(5.1-1)
Also, the value for the distance between the tines is restricted by a minimum value,
d
> 4
microns.
(5.1-2)
Since micromechanical fabrication processes are best suited for planar structures (zdimensions less than a few microns), the fabrication of the tines requires the most
complicated micromachining procedures for the gyro. Generally, precise geometries are
patterned by standard photolithographic techniques, and shallow etches cut through the
silicon wafer [23] to form a specific structure. However, a deep etch is required for
fabricating the tines. Anisotropic deep etching must be done along one of the weakly
106
bound silicon crystal planes. Generally, this requires a cut angled at about 54 degrees from
the vertical. An additional procedure to fabricate a vertical structrure is then required [22].
It is also assumed that there exists a minimum gap between the sensor wing and the
bridge electrode such that
g
>
12 microns .
(5.1-3)
The bridge electrodes are metallized onto a seperate plate which is suspended over the gyro
by bonding the plate to the silicon wafer. The sensor gap restriction is a conservative
estimate of the z-dimensional tolerance for situating the plate. Furthermore, the depth of
the strain beam is assumed equal to the depth of the sensor wing,
bz
(5.1-4)
Wz .
=
This is meant to simplify the fabrication so that seperate diffusions are not needed.
The two inertial masses consist of rectangular gold bars. Gold is chosen because of
its high density and its compatibility with the fabrication process. The gold is attached to
the tine by either metallization or electroplating [30]. For either technique, it is assumed
that the height of the gold bar is equal to the height of the tine,
mz
=
h .
(5.1-5)
Also, it is assumed that the gold can only extend from the tine in the x-direction a maximim
distancesuch that
mx
<
80 microns.
(5.1-6)
Again, these are fabrication restrictions.
The dimension representing the thickness of the flexures, fz , is not an input
quantity. Instead, the system driving frequency, which is matched to the torsional resonant
frequency, is an input. The flexure thickness is calculated from the stiffness model
presented in Section 4.3 for given values of the flexure width, fx , length, fy , and system
frequency, woO. A loop in the program iterates the flexure thickness until the desired
frequency is obtained. The system frequency is chosen as a design input instead of the
flexure thickness because it is more significant in terms of the entire system design.
107
The fabrication of the sensor wings is complicated by the requirement that the sides
of each wing are electrically isolated from each other. Electrical isolation eliminates the
need to lay wires across the flexures and wings in order to apply a differential voltage
across the tines. Instead, a differential voltage is applied across the two flexures. The
semiconductor properties of silicon allow the potential to transfer to the tines. Thin
diagonal slots are cut through the wings from a point between the tines to one side of the
flexure. This allows each flexure to be electically connected to a different tine. The two
sides of the wing are connected by a lapjoint shown in Figure 5.1. An oxide covering in a
rectangular pattern is laid over the slot. The oxide is covered by a polysilicon layer a few
microns thick which adds structural integrity to the wing. It is seen that the width of the lap
joint must be at least the width of the flexure. An apparant disadvantage of the lap joint is
that it reduces the capacitance area of the sensors. The lap joint forces an x-dimensional
gap between the sensor plates which is denoted by the variable sensgap in the program
where,
sensgap = 2 xl.
(5.1-7)
for xl defined in the sensor model of Eqn. (4.2-4). Then, assuming that the lap joint is at
least twice the flexure thickness, the variable x1 is restricted by
xl
>
fx .
(5.1-8)
The top portion of Table 5.1 lists the device geometry with the dimensions in
microns. The constrained parameters d, h, w, g, mx , and sensgap are all set at the limit
which maximizes the gyroscopic sensitivity. The tine aspect ratio is set as high as possible,
15 : 4 microns, to maximize the tine deflection per applied driving volt. The inertial mass
dimension, mx, is also set at its limit (5.1-6) to maximize the mechanical gain. The gap
between the bridge electrodes is set at its minimum value to maximize the sensor area (5.1-
8).
108
The geometric parameters that remain available for design tradeoffs include:
1
length of tines ,
my
y-dim of gold bar ,
Wx, Wy, Wz
dimensions of wings ,
fx , fy
dimensions of flexures , and
bx , by
dimensions of strain beam .
For simplicity purposes, it is also assumed that the driving frequency, driving voltage, and
electronics configuration are fixed by the system requirements. The frequency is set at
o =
2 kHz
and the differential driving voltage is set at
Vd
=
(5.1-9)
20 V.
This specific driving voltage supplies adequate motor force and is also easily compatible
with a typical motor electronics configuration. A frequency of 2 kHz is chosen since it has
been shown to be an effective frequency for the Boxenhorn micromechanical gyroscope
[2].
5.2
Performance Tradeoffs
The remaining design parameters are chosen to produce a reasonable performance
level of the gyro. The performance is defined as a balance between sensitivity and errors.
In this section, each design parameter is analyzed separately in terms of the modeled
performance characteristics that it affects.
length of tines and y-dim of mass:
109
To increase the mechanical gain of the gyro, the length of the tines and the
dimensions of the mass are made as large as possible. Long tines increase the area for the
motor electrostatic force and decrease its lateral stiffness. The mass of the gold bar also
increases the mechanical gain as shown in Eqn. (4.1-30). The foremost limiting restraint is
the estimated stability limit. For the baseline design the stability parameter is
EQ
(5.2-1)
3.16.
=
This falls within the linear response region as shown in Chapter 3.
Also, the amplitude of the tine vibration cannot exceed one-half of the gap between
the tines. A maximum value for the ratio of the tine amplitude to one-half of the tine gap is
defined as
u2
Xm
-
<
0.25.
(5.2-2)
This ratio should insure that the tines do not strike each other, and it is small enough to
satisfy the assumptions used in deriving the motor model in Chapter 4.1. Also, the tine
length must obey the constraint of Eqn. (2.22-3) which guarantees that the tine twisting
mode will not be excited. The twisting resonant frequency is listed in Table 5.1 under the
"Dynamic Characteristics" heading. For the baseline design, the frequency of the tine
twisting mode is more than a factor of six greater than system frequency.
The program variable, phi , listed under the "Motor Characteristics" heading, is
defined as the static twist angle of each tine for a z-direction gravitational acceleration. The
twist occurs because the gold bar center of mass is not aligned with the tine center of mass.
For the baseline design, my and I were chosen such that phi is less than two degrees for an
acceleration of 100 g's.
Throughout the analysis of the gyroscope, it was assumed that the natural
frequency of each tine is greater than the driving frequency. This was meant to simplify the
analysis. However, if the length of the tine is adjusted to yield a tine frequency which is
110
near the driving frequency, the mechanical gain increases per driving volt. The gain from
operating near resonance is defined as the dynamic loadfactor [9]. For the tines,
d I ftine
2
Stinfe
(5.2-3)
The baseline design capitalizes on the resonance gain to limit the necessary driving voltage.
The design fixes the tine frequency only 15% higher than the driving frequency to yield a
dynamic load factor of 4.63.
wing dimensions :
To maximize the gyro sensitivity, the wing area is made as large as possible. The
output signal is linearly related to the nominal parallel plate capacitance, CO , by Eqn. (4.23). But, increasing the sensor area has no effect on the gyro signal to noise ratio since the
capacitance constant present in the random noise equations (4.5-14) and (4.5-18) is also
linearly proportional to CO. The standard deviation of the noise thus increases at the same
rate as the sensitivity. However, increasing the length of the wing, Wx , has an added
effect on the sensitivity, besides increasing the capacitor area. Eqn. (4.2-4) indicates that
Wx
x2 = W
is a linear factor of the sensitivity. It is thus advantageous for the x-dimension
of the wing to exceed the y-dimension.
The wing dimensions strongly influence the magnitude of the air damping. For the
pumping effect of Eqn. (4.4-16), the damping force is linearly related to Wx 4 . A
successful design tradeoff between low air damping and reasonable sensitivity could not be
obtained. Instead, the baseline design assumes a vacuum package that fixes the quality
factor at Q = 200. This value was chosen because it allows a reasonable device
sensitivity,
111
eout = 0.47 millivolts per rad/sec
(5.2-4)
while maintaining a system bandwidth of 5 Hz which is significant for noise rejection. For
the baseline design dimensions the Q estimate for air damping is
Q
=
2.9 .
(5.2-5)
The damping force due to the air pumping is two orders of magnitude greater than the flow
impingement damping force, as listed in Table 5.1 under "Dynamic Characteristics" . This
means that if the gyro is to be efficiently operated in air, the sensor wings must be designed
with slits to reduce the air pumping.
flexure and strain beam dimensions :
The length and width of the flexures are chosen such that the calculated thickness
for the desired frequency is in a practical range. This generally means that the flexure
thickness is between
0.25 microns
<
<
fz
2.0 microns.
(5.2-6)
The flexure dimensions are constrained only by Eqn. (2.2-6) which guarantees that the
flexure bending will not be excited. For the baseline design, the flexure thickness is
fz
0.6 microns.
=
The strain beam dimensions are chosen to keep the maximum principle stress on the
flexures less than 1% of the yield strength. for silicon, the yield strength is assumed [25]
to be
YS
=
4.4 E9 N/m 2 .
For the baseline design, the strain beam alleviates the stress caused by a 0.2 micron boron
diffusion contraction deflection into a maximum principal stress of only 0.38% of the yield
strength.
112
to electronics
substrate
E + Vsin wt
X
2
/1
J
bridge electrode
strain beam slot
•
I
7
bridge electrode
oxide
polysilicon layer
Figure 5.1 Design Sketch
113
TUNING FORK MICROMECHANICAL GYROSCOPE PERFORMANCE
wwwwwwwwessasemovesww************************
1
d
w
h
mx
my
Wx
Wy
Wz
g
fx
fy
fz
sensgap
bx
by
bz
700.00
4.00
4.00
15.00
80.00
50.00
160.00
120.00
3.00
12.00
10.00
40.00
0.61
20.00
200.00
8.00
3.00
length of tines
gap between tines
width of tines
height of tines
width of inertial mass
length of inertial
mass
length of wings
.width of wings
height of wings
sensor gap
width of flexures
length of flexures
thickness of flexures
gap between sensor electrode plates
length of strain beam
width of strain beam
substrate thickness
microns
microns
microns
microns
microns
microns
microns
microns
microns
microns
microns
microns
microns
microns
microns
microns
microns
+++++++++++++++++MECHANICAL PROPERTIES+++++++
massg
masst
massw
masstot
Ibeamt
Ibeamw
It
Im
Iw
Io
lIox
Ioz
pmps
1.2E-0009
9.7E-0011
1.3E-0010
2.8E-0009
8. OE-0023
2.7E-0022
3.5E-0021
3.1E-0018
2.8E-0019
6.8E-0018
5.3E-0017
6.OE-0017
3.8E-0001
kg
kg
kg
kg
m^4
m^4
Kgm^2
Kgm^2
Kgm^2
Kgm^2
Kgm^2
Kgm^2
Y.
mass of each gold bar
mass of each tine
mass of each wing
total mass of structure
beam area inertia for tines
beam area inertia for wings
mass moment of inertia for tines
mass moment of inertia for gold bars
mass moment of inertia for wings
total nominal mass momt of inertia
mass momt. of inertia, x-axis
mass momt. of inertia, z-axis
percent max princ. stress on flexures
++++++++++++++++MOTOR CHARACTERISTICS+++++++
Vd
F1
F2
Fo
ym
phi
u2
11
20.00
Volts
1.2E-0006
N
3.4E-0007
N
1.5E-0006
N
.4.6E-0001 microns
2.9E-0002
degrees
2.3E-0001 unitless
1.1E-0019
kgm^2
differential driving voltage
electrostatic force, parallel plates
electrostatic force, fringing
total electrostatic force
tine midpoint drive amplitude
tine rotation angle per z-dir g
ratio of drive amplitude to maximum
variable inertia term
Table 5.1
114
++++++++++++++++DYNAMIC CHARACTERISTICS++++++++
BWol
wtineb
wwing
wtwist
wo
dlft
dlfw
dlftw
Q
Qair
Qvac
Fpu
Fpl
5.000
2.258
564.423
12.640
2.000
4.63
1.00
1.03
200.00
2.93
200.00
4.5E-0006
2.3E-0008
Hz
kHz
kHz
kHz
kHz
unitless
unitless
unitless
unitless
unitless
unitless
N
N
open loop system bandwidth
tine fundamental frequency
wing fundamental frequency
tine twisting frequency
driving frequency
dynamic load factor, tines
dynamic load factor, wings
dynamic load factor, tine twisting
system amplification factor
Q if operated at atmospheric pressure
assumed Q from vacuum chamber
damping force due to pumping
damping force due to flow around plate
++++++++++++++++++GYRO OUTPUT+++++++++++++
omega
eps
stabrat
theta
theta
thmax
ul
gtor
gtor
Co
delC
eout
1.00
1.6E-0002
0.913
1.4E-0002
0.8655
8.5960
0.0017
1.3E-0015
1.3E-0004
6.2E-0015
2.3E-0017
0.474
rad/sec
unitless
unitless
degrees
minutes
degrees
unitless
Nm
dyne-mic
farads
farads
mVolts
input rate
inertia ratio -
mechanical gain term
open loop stability ratio
output vibration amplitude
maximum vibration amplitude
ratio of output to max output
gyroscopic torque
nominal sensor capacitance
change in capacitance
output signal
++++++++++++++++++ERROR SOURCES++++++++++++++
sigbrn
sigamp
SNrat
zunb
thquadu
ulq
del
ps
2.6E-0007
2.3E-0007
956.70
0.750
0.6761
0.079
0.200
1.7E+0007
volts
volts
unitless
microns
degrees
unitless
microns
N/m^2
Brownian noise standard deviation
amplifier noise standard deviation
output signal to noise ratio
assumed tolerance of gold mass in z-dir
unbalance quadrature output
ratio of quad output to max
boron diffusion contraction on fle xure_
axial pre-stress on flexures
End of Output
Table 5.1 (continued)
I
"3
6
CONCLUSIONS
From the baseline design and performance predictions described in the previous
chapter, it is safe to conclude that the micromechanical tuning fork gyroscope merits furthur
investigation. The gyro sensitivity is shown to be at a reasonable level relative to the
possible applications. The output voltage is predicted to be
eout
0.47 millivolts per rad/sec .
=
None of the identified error sources appear to fundamentally limit the effectiveness
of the gyro as an angular rate sensor. By modeling the effect of the error sources, the
output signal to noise ratio is predicted at
S/N
= 957
which translates to a minimum detectable rate of
Omin
=
218 deg/hr.
Originally, the biggest question concerning the feasibility of the device was whether
the periodic terms present in the equation of motion would lead the system response to
become nonlinear. The high frequency and low damping necessary for a micromechanical
instrument increase the significance of the periodic terms.
This question was adressed in Chapter 2 and it was shown that the response is
linear and stable within a specified parameter region. A sufficient condition for stability
was determined by transforming the equation of motion to the Mathieu equation. The
stability condition is
eQ
< f-2
where,
10
Q a system quality factor.
116
The stability was checked by applying Floquet Theory to the LP equation of motion. The
eigenvalues of the system discrete state transition matrix were shown numerically to lie
within the complex unit circle for the region specified above.
If the periodic terms in the equation of motion are neglected, the output of the
gyroscope, written in terms of the rotation angle, is a linear function of the input rate
0 (t) =
--Q
sin cot .
With the periodic terms included, the system response was approximated by a Fourier
series. It was shown that the linear approximation above is valid for the region
eQ < 4.
For values of the stability parameter ( eQ) exceeding four, an output phase shift occurs, the
first sine harmonic component decreases, and higher harmonic terms become more
significant. The linear response within the specified region was further verified by
numerical simulation.
The next feasibility concern stemmed from the demonstration model. It questioned
whether undesirable vibration modes would be excited. From a model of the mechanics, it
was shown that the device is easily designed to be rigid. The most significant mode is
twisting of each tine about its longitudinal axis. For the baseline design, the frequency of
this mode was shown to be more than a factor of six greater than the system frequency.
Brownian noise is a problem that plagues the design of any micromechanical
instrument. The low energy levels of micromechanical instruments can lead to Brownian
noise levels that restrict instrument effectiveness. However, for the baseline design
presented in this paper, the error due to the Brownian noise is on the same order as the
amplifier noise. Hence, Brownian noise is not a dominating error source. In fact, none of
the identified error sources are shown to dominate. These include cross-axis sensitivity,
external acceleration and vibration, motor-sensor coupling, and unbalance torques.
117
The only major disappointment from the performance predictions is the requirement
of a vacuum package. It was previously believed that the air damping for a tuning fork
configuration might be low enough to eliminate the need of a controlled pressure chamber.
However, the air flow through the channel defined by each sensor wing and its bridge
electrode, which is pumped by the oscillating wing, creates a substantial pressure drop
across each wing. Therefore, unless a specific damping reduction design is implemented,
the tuning fork gyro will be most efficiently operated in a vacuum. The baseline design
study shows that a moderate vacuum quality factor of Q = 200 will yield reasonable
sensitivity and open loop bandwidth.
The baseline design bandwidth of 5 Hz is obviously not adequate for many angular
rate sensor applications. This implies that the gyro must be operated with a closed-loop
rebalance servo to satisfy the requirements of some possible applications. Then the system
designer could choose the closed-loop bandwidth depending on the requirements of the
particular application. In this paper, the gyro is studied strictly from an open-loop
standpoint. The closed-loop design is left as future research.
In order to put the gyro performance predictions in their proper perspective,
comparisons to other angular rate sensors are made. First, the tuning fork configuration is
compared to Boxenhorn's double gimbal configuration [1,2]. The gyro is then compared
to previously developed tuning fork gyroscopes. The comparison is presented in an
attempt to convince the reader that a tuning fork gyroscope on a micromechanical level has
success potential, even though many conventional scale tuning fork gyroscope
development efforts have failed.
6.1
Comparison with Current Micromechanical Gyro Configuration
Table 6.1 lists various characteristics of the current double gimbal micromechanical
configuration next to the corresponding tuning fork characteristics. The original goal of the
118
tuning fork study was to determine if the configuration can give performance levels
competitive with the current configuration. The unique property of the tuning fork
configuration is that the input axis is in the same plane as the gyro chip. Therefore, if a
competitive performance level is obtained, the two micromechanical configurations can be
combined for various system applications.
As seen from the table, the two configurations have similar characteristics. Yet, it
is seen that in a few regards the tuning fork configuration has properties which are more
favorable than the current configuration. The observed favorable and unfavorable
comparisons are listed and described below:
Favorable Properties of Tuning Fork Configuration:
1)
More gyroscopic torque is possible. The gyroscopic torque of the current
configuration is a function of the driver angle which is limited to very small
values.
2)
Motor-sensor coupling is reduced. No electric flux from the tuning fork motor
crosses the resonator gap.
3)
Only one set of flexures is needed. This increases the reliability. Also, the
second set of flexures can lead to high stress levels and buckling from boron
diffusion deflections.
4)
No sticking from a rotational motor. The lateral motor of the tuning fork
eliminates the problem of sticking from motor startup instability.
119
Unfavorable Properties of Tuning Fork Configuration:
1)
The motor is nonlinear. For an effective device, the motor deflection must be
sensed and driven simultaneously in an appropriate scheme to linearize the
relationship between the mechanical gain and the driving voltage.
2)
Difficult fabrication of vertical tines. Unique fabrication processes are required
to make the tines vertical with a high aspect ratio.
3)
Smaller sensor area. A gimballed structure has more available area for the
sensor plates.
4)
The tuning fork configuration is unproven. No testing has been performed.
It is possible that further research could eliminate the unfavorable characteristics.
The most important favorable characteristic is the high gyroscopic torque. As seen from
Table 6.1 the tuning fork gyroscopic torque is more than one order of magnitude greater
than the current configuration. This is an advantage of the lateral electorstatic motor which
was extensively modeled in Chapter 4.1. High gyroscopic torque is essential to increase
the output signal-to-noise ratio.
6.2
Comparison with Previously Developed Tuning Fork Gyros
Table 6.2 lists the characteristics of two previously developed tuning fork
gyroscopes. The Sperry Gyrotron was introduced more than 30 years ago and is not still
available. Its characteristics as listed are obtained from Reference [3] . The other tuning
fork gyroscope listed is the Watson Angular Rate Sensor (ARS). The ARS is a solid state
120
device which is currently commercially available, and its characteristics are obtained from a
manufacturer specification sheet.
The primary differences between these two devices and the micromechanical tuning
fork gyro are size and power. Also, it must be noted that both of these devices require that
the tine natural frequency is tuned to the torsional natural frequency. This requires
precision testing on each individual device. A list of favorable and unfavorable
comparisons between the micromechanical tuning fork gyro and the other tuning fork gyros
is shown below:
Favorable Properties of Micromechanical Tuning Fork :
1)
High frequency. The micromechanical tuning fork gyro is routinely driven at
frequencies between 1 and 5 khz which subdues zero rate errors.
2)
Constant damping and temperature packaging more feasible due to small size.
The system characteristics are more easily controlled in a small package.
3)
Torsional to lateral tuning not necessary. Tuning the tine frequency to the
torsional frequency increases the cost and manufacturing complexity. The
micromechanical configuration does not require this.
4)
No joints or bearings. This reduces hysterises effects, structural damping, and
stress concentration areas.
5)
Small fabrication tolerances. Microfabrication techniques produce precise
dimensions. Symmetrical devices are easily produced, and unbalance and
misalignment effects are minimized.
6)
Low cost / small size. The bottom line is that the large scale tuning fork
gyroscope is not successful because it cannot compete with high accuracy,
121
conventional gyroscopes. However, the micromechanical tuning fork
gyroscope attacks a different market. The unique qualities of a micromechanical
instrument open up applications where only moderate accuracy is needed.
Future Research
6.3
The author hopes that this paper will inspire a continued study into the design of a
micromechanical tuning fork gyroscope. A long road lies ahead, but it is the resounding
conclusion that a micromechanical tuning fork gyroscope could be extremely successful.
Throughout the world, micromechanical engineering is an active research topic. The
efficiency of miicromechanical fabrication and testing procedures is increasing
continuously. It is possible to forecast that soon into the next century the sensor field will
be dominated by micromechanical instruments.
A few areas that would make interesting research topics in a continued study of the
micromechanical tuning fork gyroscope are mentioned here. First, it is necessary to
linearize the motor. This would require an extensive model of the electrostatics and a
design scheme that would sense the tine deflection while simultaneously driving the tines.
Another possible research area is a fluid mechanics study to model the quality factor of the
resonator. Because the system Q is an important design parameter, it would be helpful to
develop methods to predict and regulate the damping. The mechanical properties of the
device are also largely unexplored. The stiffness properties of the flexures could be
investigated in terms of thermal effects, fatigue effects, and residual stresses from the
fabrication processes.
,
A continued design study could lead to prototype fabrication and testing. Then,
further down the road, a closed loop design for the device could be investigated followed
by a complete inertial system study.
122
COMPARISON OF MICROMECHANICAL TUNING FORK
GYRO DESIGN TO CURRENT MICROGYRO
TUNING FORK
CURRENT MICROGYRO
CONFIGURATION
single gimbal, closed fork
double gimbal, vibratory
MOTOR
electrostatic, lateral (nonlinear)
electrostatic, rotational
SENSOR
variable capdcitance
variable capacitance
INPUT AXIS
in-plane, fork stem
normal to plane
MATERIAL
silicon chip
silicon chip
PACKAGING
temp. controlled, vacuum
temp. controlled, vacuum
SIZE
4.9 E-12 m^3
8.6 E-12 m^3
MASS
2.8 E-9 kg
9.41 E-9 kg
POWER
order of 1.0 E-12 W
order of 1.0 E-12 W
FREQUENCY
2 kHz
2.5 kHz
Q
200
780
RESONANCE GAIN
4.63
3.55
SYSTEM BANDWIDTH
5 Hz
3.2 Hz
SENSITIVITY
0.474 mV per rad/sec
0.288 mV per rad/sec
DRIVING VOLTAGE
20 V
20 V
FLEXURE THICKNESS
0.61 microns
0.48, 0.95 microns
% MAX PRINC. STRESS
0.19%
0.15%
RMS NOISE
0.5 microvolts
31 microvolts
ERROR RATE
218 deg/hr (predicted)
22,000 deg/hr (demonstrated)
GYRO TORQUE
1.3 E-4 dyne-micron
8.2 E-6 dyne-micron
STABILITY RATIO
0.91
0.007
SENSOR CAPACITANCE
9.3 E-15 farads
1.8 E-14 farads
Table 6.1
123
CHARACTERISTICS OF PREVIOUSLY DEVELOPED
TUNING FORK GYROSCOPES
SPERRY GYROTRON
WATSON ARS
CONFIGURATION
single gimbal, open fork
no gimbal, open fork
MOTOR
electrostatic plates (linear)
piezoelectric beams
SENSOR
electromagnetic pickoffs
piezoelectric beams
INPUT AXIS
in-plane, fork stem
in-plane, fork stem
MATERIAL
machined 1%carbon steel
solid state
PACKAGING
controlled damping vanes
solid state packaging, air
SIZE
1.58 E-3 m^3
6.27 E-5 m^3
MASS
7.0 kg
0.11 kg
POWER
0.5 W
0.6 W
FREQUENCY
940 Hz
360 Hz
Q
variable
not specified
RESONANCE GAIN
tuned to resonance
tuned to resonance
SYSTEM BANDWIDTH
11.76 Hz (closed loop)
50 Hz (closed loop)
SENSITIVITY
433.16 mV per rad/sec
19.1 V per rad/sec
MAX RMS NOISE
50 microvolts
15 millivolts
DRIVING VOLTAGE
unspecified
unspecified
ERROR RATE
23.8 deg/hr
162 deg/hr
STABILITY RATIO
none indicated
none indicated
Table 6.2
124
REFERENCES
[1]
Boxenhorn, B. , The MicromechanicalPlanarGyroscope Summary Report, Fiscal
83-84, IR&D Project #193, Publication CSDL-D-5734, 1984.
[2]
Boxenhorn, B. & Greiff, P. ," A Vibratory Micromechanical Gyroscope ", AIAA
Guidance,Navigation, & Control Conference, Minneapolis, MN, Aug. 1988.
[3]
Barnaby, R.E., Chatterton, J.B., Gerring, F.H. , " General Theory and
Operational Characteristics of the Gyrotron Angular Rate Tachometer",
AeronauticalEngineeringReview, Nov. 1953.
[4]
Hunt, G., & Hobbs, O. , " Development of an Accurate Tuning Fork Gyroscope",
Proceedingsof Instrumentationand MechanicalEngr. ,Vol. 179, Pt 3E, 1964.
[5]
Fearnside, K., & Briggs, B. , " The Mathematical Theory of Vibratory Angular
Rate Tachometers", The Institute of ElectricalEngineering, Monograph No.
264M, Nov. 1957.
[6]
Lyman, J., "A New Space Rate Rate Sensing Instrument" , Aeronautical
EngineeringReview, Nov. 1953.
[7]
Roark, R.J. & Young, W.C., Formulasfor Stress and Strain, McGraw-Hill
Book Co,New York, 1975,
[8]
Lang, H.L, & Bart, S.F. , " Toward the Design of Successful Electric
Mocromotors", Proceedingsof the IEEE Workshop on Microrobotsand
Teleoperators, 1988.
[9]
Thomson, W. T., Theory of Vibration with Applications, Prentice-Hall, Inc. ,
Englewood Cliffs, NJ, 1981.
[10]
Crandall, S., Karnopp, D., Kurtz, E., & Pridmore-Brown, D., Dynamics of
Mechanical and ElectromechanicalSystems, D. E. Krueger Publishing Company,
New York, 1984.
[11]
Richards, J.A., Analysis of PeriodicallyTime-Varying Systems, Springer-Verlag,
New York, 1983.
[12]
McLachlan, W., Theory & Application of Mathieu Functions, Clarendom Press,
Oxford, 1947.
[13]
Porter, B., Stability Criteriaof LinearDynamical Systems, Oliver & Boyd,
London, 1967.
[14]
Battin, R.H. , An Introduction to the Mathematics and Methods of Astrodynamics,
AIAA Education Series, 1987.
[15]
Smythe, W.R., Static & Dynamic Electricity, McGraw-Hill Book Co., New York,
1968.
125
[16]
Gibbs, W.J., Conformal Transformationsin ElectricalEngineering , Chapman &
Hall, London, 1958.
[17]
Bowman, F., Introduction to Elliptic Functionswith Applications , Dover, New
York, 1961.
[18]
Churchill, R.V., OperationalMathematics , McGraw-Hill Book Co., 1972.
[19]
Press, W.H., Flannery, B.P., Teukolsky, S.A., & Vetterling, W.T., Numerical
Recipes - The Art of Scientific Computing , Cambridge University Press, New
York, 1986.
[20]
Shigley, J.E., & Mitchell, L.D., MechanicalEngineering Design , McGraw-Hill
Book Co., New York, 1983.
[21]
Timoshenko, S.P., & Goodier, J.N., Theory of Elasticity , McGraw-Hill Book
Co., New York, 1970.
[22]
Kaminsky, G., " Micromachining of Silicon Mechanical Structures ", Journalof
Vacuum Science Technology, July/August 1985.
[23]
Peterson, K.E., " Silicon as a Mechanical Material ", Proceedings of the IEEE,
Vol. 70, No. 5, May 1982.
[24]
Connally, J.A., " Torsion of a Thin Rectangular Beam with Axial Prestress and
Ends Constrained from Warping ", Master of Science in Mechanical Engineering
Thesis, MIT, 1987.
[25]
Wolf, H. , Silicon Semiconductor Technology, John Wiley & Sons, Inc., New
York, 1971.
[26]
Potter, M.C. , & Foss, J.F., Fluid Mechanics , Great Lakes Press, Inc., Okemos,
MI, 1982.
[27]
Happel, J., Low Reynold's Number Hydrodynamics with Special Applications to
ParticulateMedia , Prentice-Hall, Englewood Cliffs, NJ, 1965.
[28]
Brown, R.G., Random Signal Analysis and Kalman Filtering , John Wiley &
Sons, Inc., New York, 1983.
[29]
Newton, G., " Comparison of Vibratory & Rotating Wheel Gyroscopic Rate
Indicators ", Proceedingsof the AIEE, July 1960.
[30]
Terry, S., Angell, J., & Barth, P., " Silicon Micromechanical Devices ", Scientific
American , April 1983.
[31]
Findlay, W., & Watt, D., PASCAL: An Introduction to Methodical Programming,
Computer Science Piess, Inc., Rockville, MD, 1978.
[32]
Morrow, C.T., " Zero Signals in the Sperry Tuning Fork Gyrotron ", Journalof
the Acoustical Society of America , Vol. 27, No. 3, May 1955.
[33]
Ogata, K., Modern Control Engineering , Prentice-Hall ,Inc., Englewood Cliffs,
NJ, 1970.
126
APPENDIX
127
(Enable 8087 coprocessor)
(Turn on user interrupt)
(SN+)
($I+)
program simulation;
(This is a PASCAL program which uses fourth order Runge-Kutta integration
with error monitored adaptive step size that solves the tuning fork gyro
periodically forced time-variant equation of motion. The second order
differential equation used assumes a tuned device so that the driving
frequency is equal to the torsional resonant frequency. The oscillatory
output is calculatedand values greater than a specified transient time
are stored and printed in tabular and graphical form. The system
parameters are inputted interactively.)
CONST
n = 2;
nstep = 200;
(# of states)
(max # of stored steps)
TYPE
glnarray = ARRAYC1..n3 of DOUBLE;
glarray
(dep. variable array)
= glnarray;
VAR
: ARRAYE1..nstep3 of DOUBLE;
xp
ystart,dydxyout: glnarray;
yp : ARRAY[1..n,1..nstep3 of DOUBLE;
(gyro inputted parameters)
woQ,eprate: DOUBLE;
xlx2,xtrans: DOUBLE;
thmax,pernper ypmax,ratio: DOUBLE;
(adaptive algorithm parameters)
eps,hlhmindxsav: DOUBLE;
(counters)
i,kmaxkount,nbadnok:INTEGER;
-------------------------------------------------------------------------(this procedure defines the system
parameters)
PROCEDURE start;
Begin
writeln('input drive frequency in rad/sec');
readln(wo);
writeln('input ep, inertia ratio times drive angle in rad');
readln(ep)4
writeln('input quality factor, Q');
readln(Q);
writeln('input applied rate in rad/sec');
readln(rate);
writeln('input # of periods for stored values');
readln(nper);
writeln('input allowed transient time in seconds');
readln(xtrans);
End;
------------------------------------------------------------------------(this procedure defines the state
space equations integrated by RK4)
1I18
PROCEDURE derivs(x:double; y:glnarray;
VAR
dydx:glnarray);
Begin
dydx[ 13
dydx[23
y123;
((-wo/Q -ep*wo*COS(wo*x))*y[23 -wo*wo*y[13
-rate*ep*wo*COS(wo*x))/(l+ep*SIN(wo*x));
End;
C
---------------------------------------------------------------------------(this procedure computes the numerical
integration using 4th order Runge-Kutta
(rk4) and the given state equations
(derivs) for a given step size.)
PROCEDURE rk4(ydydx: g1narray; n: integer; x,h: real; VAR yout: glnarray);
VAR
i: integer;
xh,hhh6: real;
dymdytyt: glnarray;
BEGIN
hh := h*O.5;
h6 := h/6.0;
xh := x+hh;
FOR i := 1 to n DO BEGIN
ytCi3 := yi3]+hh*dydxi3]
END;
derivs(xh ,yt,dyt);
FOR i := 1 to n DO BEGIN
ytCi3 := y~i3+hh*dyti3]
END;
derivs(xh,yt,dym);
FOR i := 1 to n DO BEGIN
yt[i3 := yti3+h*dymCi3;
dymEi3 := dytli3+dymti3
END;
derivs(x+hytdyt);
FOR i := 1 to n DO BEGIN
youtti3
(first
Euler
step @ midpoint)
(2nd Euler step a midpoint w/ new deriv.)
(Euler step Z endpoint)
(final integr. estimate over interval
from Runge-Kutta algorithm)
:= yCi3+h6*(dydxCEi+dytEi3+2.0*dymE3])
END
END;
C------------------------------------------------------------------------{ This Procedure calculates the stepsize
bounded by the error criterion given a trial
value and the current values of the
arrays )
PROCEDURE rkqc(VAR y,dydx: glarray; n:
integer; VAR x: real;
129
htry,eps: real; yscal: glarray; VAR hdid,hnext: real);
LABEL 1;
CONST
pgrow=-0.20;
exp. if desired accuracy < calculated }
pshrnk=-0.25;
exp. if desired accuracy > calculated )
fcor=0.08666666;
1.0/15.0 )
one=1.0;
safety=0.9;
errcon=6.0e-4;
( 4/safety
*
1/pgrow
>
VAR
i: integer;
xsav,hh,h,temp,errmax: real;
dysav,ysav,ytemp: glarray;
BEGIN
:= x;
C save initial values )
FOR i := 1 to n DO BEGIN
ysavli3 := yti3;
dysavCi3 := dydxti3
END;
h := htry;
( initialize stepsize @ trial
value )
1:
hh := 0.5*h;
C take 2 half steps and comp. num. estimate }
rk4(ysav,dysav,n,xsav,hh,ytemp);
x := xsav+hh;
derivs(x,ytemp,dydx);
rk4(ytemp,dydx,n,x,hh,y);
C take 1 large step and comp. num. estimate }
x := xsav+h;
IF (x = xsav) THEN BEGIN
C check that stepsize is not infinitesimal )
writeln('pause in routine RKQC');
writeln('stepsize too small'); readln
END;
rk4(ysav,dysav,n,xsav,h,ytemp);
errmax
O.0;
0=
FOR i := 1 to n DO BEGIN
C truncation error is comparison of 2 estims.)
ytempEi) := yCi]-ytempi3];
C error is scaled by estimated output value )
temp := abs(ytempti3/yscalti]);
IF (errmax < temp) THEN errmax := temp
END;
errmax := errmax/eps;
{ error relative to tolerance, eps )
IF (errmax > one) THEN BEGIN { too large - shrink stepsize )
h := safety*h*exp(pshrnk*ln(errmax));
GOTO 1 END
C try again with new stepsize )
ELSE BEGIN
xsav
hdid
:= h;
IF (errmax > errcon) THEN BEGIN
{ error below tolerance - increase stepsize }
hnext := safety*h*exp(pgrow*ln(errmax))
END ELSE BEGIN
hnext := 4.0*h
END
END;
FOR i := 1 to n DO BEGIN
C add fifth order truncation error )
yCi] := yEi3+ytempti3*fcor
END
END;
130
-------------------------------------------------------------------------
{ This procedure steps the numerical
integration storing values of the
output array as specified
PROCEDURE odeint(VAR ystart: glnarray; nvar: integer;
xl,x2,eps,hl,hmin: real; VAR nok,nbad: integer);
LABEL 99;
CONST
maxstp=10000;
{ max allowable steps for computer time
two=2.0;
zero=0.0;
tiny=1.0e-30;
VAR
nstp,i: integer;
xsav,x,hnext,hdid,h: real;
yscal,y,dydx: glnarray;
BEGIN
x := xl;1
{ initialize time variable )
IF (x2 > xl) THEN h := abs(hl ) ELSE h := -abs(hl);
nok := O;
{ initialize counters )
nbad := O;
kount
O;
0=
FOR i := 1 to nvar DO BEGIN
yi)] := ystart[i3
END;
xsav := x-dxsav*two;
FOR nstp := 1 to maxstp DO BEGIN
derivs(x,y,dydx);
FOR i := 1 to nvar DO BEGIN
C estim. output val. used to scale error
yscali3 := abs(y[i3)+abs(dydxti)*h)+tiny
END;
IF
(x
>
xtrans)
THEN
BEGIN
(
only
(
IF
only
store
store
after
if
transient
interval>dxsa
(abs(x-xsav) > abs(dxsav)) THEN BEGIN
C only store if #steps < kmax
IF (kount < kmax-1) THEN BEGIN
kount := kount+1;
xp[kount3 := x;
FOR i := 1 to nvar DO BEGIN
yp[i,kount] := yti3
END;
xsav := x
END
END
END;
C if remaining time < stepsize }
IF (((x+h-x2)*(x+h-xl)) > zero) THEN h := x2-x;
rkqc(y,dydx,nvar,x,h,eps,yscal,hdid,hnext);
IF (hdid = h) THEN BEGIN
{ count successful steps >
nok := nok+1
131
)
)
END ELSE BEGI N
nbad : =-nb ad+l
( count retried steps }
END;
( check if integration
(((x-x2)*( x2-xl)) >= zero) THEN BEGIN
FOR i := 1 to nvar DO BEGIN
:= yi3]
ystart[ i3
END;
IF (kmax <> O) THEN BEGIN
kount := kount+1;
xptkount] := x;
( save final step )
FOR i := 1 to nvar DO BEGIN
yp[i,kount] := yi3]
END
END;
GOTO 99
( go to end }
END;
h := hnext;
( return stepsize
END;
99:
END;
interval completed
IF
------------------------------------------------------------------------ >
{ This procedure writes the system parameters
and tabulated stored output angles to the
file OUTPT )
PROCEDURE writeout;
0
outpt: TEXT;
i,j: INTEGER;
Begin
assign(outpt,' prn');
rewrite(outpt)
writeln(outpt,
RESIIL .TS ');
TUNING FORK MICROGYRO SIMULATION
writeln(outpt)
writeln(outpt)
writeln(outpt,
writeln(outpt,
writeln(outpt,
Inputted Parameter Values'
'
~-------------------------'
w =
',wo:9:4,'
rad/sec
','
',ep*Q:B:2);
writeln(outpt, 'input rate = ',rate:9:2,' rad/sec','
nper = ',nper:9:2,'
writeln(outpt, •
,
xtrans:9:2,' seconds');
Q = ',Q:9:2,
P ep*Q =
ep
= ',ep:9:6);
xtrans =
writeln(outpt)
writeln(outpt,
writeln(outpt,
time (sec)','
','output angle (rad)',
angular velocity (rad/sec)');
--------------------------
writeln(outpt);
for i := 1 to kount do
begin
writeln(outpt,xpti]:12:6,'
-)
',yp
13Z
1, i3:14:6,
',yp[2,i3:16:6);
end;
writeln(outpt);
writeln(outpt,' estimated max output angle
writeln(outpt,' period of each output cycle
# of successful steps
writeln(outpt,'
# of bad steps
writeln(outpt,'
writeln(outpt,' simulated max output angle
writeln(outpt,' ratio simulated/estimated
writeln(outpt);
close(outpt);
End;
',thmax:14:6,' radians');
',per:14:5,' seconds');
',nok:14);
',nbad:14);
',ypmax:14:6,' radians');
',ratio:14:5);
(---------------------------------------------------------------------------
BEGIN
(Main Program)
ystart[13 := 0.0;
ystart[23 := 0.0;
(initial
start;
(input parameters)
per := 6.28/wo;
xl := 0.0;
x2 := nper*per+xtrans;
(cycle period)
(start at time zero)
(integrate until inputted transient plus
desired number of stored periods)
thmax := Q*ep*rate/wo;
(predicted maximum output angle)
values)
dxsav := (x2-xtrans)/(20.0*nper );
{store 20 values per period)
(comp. values accurate within 0.1%)
eps := 1.0e-3;
(max # of values that can be stored)
kmax := 120;
(minimum allowable stepsize)
hmin := 0.0;
(initial stepsize 15 per period)
hi := per/15;
odeint(ystart,n,xl,x2,eps,hl,hmin,nok,nbad);
(drive num. integration)
(scan stored output for max)
ypmax := ABS(yp[1,13);
FOR i := 2 to kount DO BEGIN
IF(ABS(yp1l,i3) > ABS(ypmax)) THEN ypmax:=ABS(yp[1,i3);
END;
ratio := ypmax/thmax;
writeout;
END.
(Main Program)
133
TUNING FORK MICROGYRO SIMULATION RESULTS
Inputted Parameter Values
input
time
w = 6280.0000 rad/sec
Q
rate =
1.00 rad/sec
ep
nper =
4.00
xtrans
(sec)
output angle (rad)
0.600073
0.600161
0.600260
0.600352
0.600443
0.600547
0.600640
0.600735
0.600824
0.600916
0.601009
0.601105
0.601194
0.601289
0.601378
0.601477
0.601565
0.601653
0.601750
0.601837
0.601927
0.602020
0.602115
0.602206
0.602301
0.602389
0.602490
0.602576
0.602663
0.602762
0.602851
0.602941
0.603035
0.603129
0.603222
0.603315
0.603402
0.603508
0.603606
0.603696
0.603799
0.603898
0.603990
0.604000
- - •i F
=;."III-
-7.883974
-13.806398
-14.742356
-10.611256
-3.653795
4.929047
11.206867
14.400010
13.219787
7.474279
-1.552215
-10.453149
-14.819492
-13.894013
-8.825099
-0.874646
6.246314
11.830887
14.506721
12.701519
6.565329
-2.699352
-11.178645
-15.014945
-13.454859
-8.088356
0.206513
7.077917
12.294250
14.515240
12.078335
5.346603
-4.160377
-12.038132
-15.131368
-12.793577
-7.098503
1.613911
9.091112
13.515865
14.054310
9.052120
0.569516
-0.494348
0.000074
0.001296
0.002187
0.002378
0.001695
0.000463
-0.000919
-0.001890
-0.002297
-0.001976
-0.001117
0.000239
0.001454
0.002267
0.002328
0.001544
0.000233
-0.001106
-0.001988
-0.002282
-0.001739
-0.000705
0.000765
0.001951
0.002412
0.002412
=
/
=+
;
-
=
=
=
=
16.00
angular velocity (rad/sec)
0.002091
0.001106
-0.000363
-0.001552
-0.002218
-0.002147
-0.001376
-0.000129
0.001136
0.002113
0.002402
0.001803
0.000639
-0.000770
-0.001805
-0.002293
-0.002051
-0.001236
estimated max output angle
period of each output cycle
# of successful steps
# of bad steps
simulated max output angle
•
200.00 ep*Q =
0.080000
0.60 seconds
0.002548 radians
0.00100 seconds
5461
1061
0.002412 radians
f*
QIL•
~
13i-
TUNING FORK MICROGYRO SIMULATION RESULTS
Inputted Parameter Values
Q
6280.0000 rad/sec
1.00 rad/sec
ep
input rate =
xtrans
4.00
nper =
w
time
(sec)
0.600088
0.600182
0.600275
0.600374
0.600465
0
*
0.600564
0.600662
0.600750
0.600853
0.600948
0.601041
0.601129
0.601217
0.601315
0.601411
0.601501
0.601605
0.601699
0.601793
0.601881
0.601969
0.602065
0.602162
0.602250
0.602352
0.602446
0.602541
0.602629
0.602717
0.602815
0.602902
0.602990
0.603089
0.603184
0.603276
0.603376
0.603466
0.603565
0.603663
0.603752
0.603855
0.603949
0.604000
output angle
(rad)
0.000627
0.000432
0.000099
-0.000289
-0.000548
-0.000634
-0.000490
-0.000203
0.000203
0.000512
0.000644
0.000568
0.000321
-0.000060
-0.000410
-0.000605
-0.000601
-0.000385
-0.000035
0.000304
0.000559
0.000643
0.000494
0.000199
-0.000205
-0.000503
-0.000635
-0.000564
-0.000326
0.000047
0.000377
0.000596
0.000627
0.000432
0.000099
-0.000289
-0.000548
-0.000634
-0.000490
-0.000203
0.000203
0.000512
0.000610
estimated max output angle
period of each output cycle
# of successful steps
# of bad steps
simulated max output angle
ratio simulated/estimated
=
=
=
=
=
=
200.00 ep*Q =
0.020000
0.60 seconds
angular velocity
4.00
(rad/sec)
-0.954790
-3.022520
-3.989034
-3.556727
-2.022114
0.325526
2.532851
3.790011
3.839891
2.478833
0.264849
-1.933476
-3.523502
-3.999616
-3.045972
-1.222964
1.302022
3.172200
4.012027
3.577275
2.042057
-0.340796
-2.623704
-3.851034
-3.788681
-2.429728
-0.249349
1.839262
3.423731
4.017464
3.299185
1.557704
-0.954794
-3.022532
-3.989050
-3.556740
-2.022122
0.325527
2.532860
3.790025
3.839906
2.478843
1.333530
0.000637 radians
0.00100 seconds
5640
812
0.000644 radians
1.01054
I ?C
TUNING FORK MICROGYRO SIMULATION RESULTS
Inputted Parameter Values
w
6280.0000 rad/sec
0
rad/sec
1.00
ep
xtrans =
4.00
=
input rate =
nper =
time
(sec)
output angle
0.600047
0.600134
0.600222
0.600320
0.600417
0.600506
0.600610
0.600704
0.600799
0.600886
0.600973
0.601071
0.601168
0.601257
0.601360
0.601454
0.601549
0.601636
0.601724
0.601822
0.601919
0.602008
0.602111
0.602205
0.602300
0.602387
0.602474
0.602573
0.602670
0.602759
0.602863
0.602957
0.603051
0.603138
0.603225
0.603324
0.603421
0.603510
0.603614
0.603707
0.603803
0.603889
0.603976
0.604000
#
of
bad
angular velocity
0.000160
0.000141
0.000079
-0.000016
-0.000104
-0.000153
-0.000150
-0.000095
-0.000006
0.000078
0.000140
0.000160
0.000122
0.000047
-0.000055
-0.000129
-0.000160
-0.000140
-0.000079
0.000016
0.000105
0.000153
0.000151
0.000095
0.000005
-0.000078
-0.000140
-0.000159
-0.000121
-0.000046
0.000056
0.000130
0.000160
0.000139
0.000077
-0.000018
-0.000106
-0.000153
-0.000150
-0.000093
-0.000004
0.000079
0.000140
0.000150
estimated max output angle =
period of each output cycle =
# of successful steps
=
*
(rad)
200.00 ep*Q =
0.005000
0.60 seconds
steps
=
simulated max output angle =
ratio simulated/estimated
=
0.047323
-0.486915
-0.877212
-1.000792
-0.761647
-0.299180
0.342234
0.808772
1.005337
0.881311
0.495664
-0.101241
-0.659535
-0.962838
-0.943603
-0.592941
-0.036354
0.488220
0.874454
1.001545
0.762074
0.295211
-0.348116
-0.815172
-1.005825
-0.874823
-0.489044
0.107688
0.659254
0.961853
0.943964
0.589314
0.027827
-0.500638
-0.883193
-0.999205
-0.753023
-0.284482
0.354912
0.815535
1.005831
0.876112
0.488425
0.350944
0.000159 radians
0.00100 seconds
5507
958
0.000160 radians
1.0075L
1.00
(rad/sec)